Functional Analysis and Continuous Optimization: In Honour of Juan Carlos Ferrando's 65th Birthday, Elche, Spain, June 16–17, 2022 3031300130, 9783031300134

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Springer Proceedings in Mathematics & Statistics

José M. Amigó María J. Cánovas Marco A. López-Cerdá Manuel López-Pellicer   Editors

Functional Analysis and Continuous Optimization In Honour of Juan Carlos Ferrando’s 65th Birthday, Elche, Spain, June 16–17, 2022

Springer Proceedings in Mathematics & Statistics Volume 424

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

José M. Amigó · María J. Cánovas · Marco A. López-Cerdá · Manuel López-Pellicer Editors

Functional Analysis and Continuous Optimization In Honour of Juan Carlos Ferrando’s 65th Birthday, Elche, Spain, June 16–17, 2022

Editors José M. Amigó Center of Operations Research Miguel Hernández University of Elche Elche, Alicante, Spain Marco A. López-Cerdá Department of Mathematics Alicante University San Vicente del Raspeig, Alicante, Spain

María J. Cánovas Center of Operations Research Miguel Hernández University of Elche Elche, Alicante, Spain Manuel López-Pellicer Institute for Pure and Applied Mathematics (IUMPA) Polytechnic University of Valencia Valencia, Spain

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-031-30013-4 ISBN 978-3-031-30014-1 (eBook) https://doi.org/10.1007/978-3-031-30014-1 Mathematics Subject Classification: 03G10, 06A11, 20K15, 20K25, 20K40, 20K45, 22A05, 22B05, 22D05, 22D10, 22D35, 30H20, 43A40, 46A03, 46A04, 46A08, 46B10, 46B20, 46E05, 46E10, 46E15, 46E25, 46B26, 46S10, 47A35, 47B33, 47S10, 54C35, 54C40, 54E52, 54H11 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is the outgrowth of the International Meeting on Functional Analysis and Continuous Optimization (IMFACO) that was held in Elche (Spain) on June 16 and 17, 2022 in honour of Prof. Juan Carlos Ferrando on the occasion of his 65th birthday. Since its inception, this meeting was organized to be a forum where experts in Functional Analysis and Continuous Optimization could present their results, exchange ideas and discuss topics of common interest, thus creating synergies between both disciplines. The twelve chapters of this book comprise communications presented at IMFACO on current research topics in Functional Analysis and Continuous Optimization, namely, Banach spaces, degree of non-densifiability, function spaces, linear semi-infinite optimization, multiobjective optimization, non-Archimedean fields, subdifferentiability, topological Abelian groups, variational convexity and vector lattices. In addition, the first chapter summarizes the mathematical work of Prof. Juan Carlos Ferrando. It is worth mentioning that several of the new results presented in these chapters have been made possible thanks to discussions at IMFACO between researchers of both disciplines. We hope that the readers will find helpful information and even inspiration for their research. Furthermore, this book is the result of a collective effort. The editors are very grateful to Dr. Francesca Bonadei, Banu Dhayalan and Gowtham Chakravarthy V. (Springer Nature) for their guidance and assistance during the publication process, and to the authors for their excellent work and timely submission. We also thank the Scientific and Organizing Committees of IMFACO, the invited speakers and all the participants for their dedication and enthusiasm. Finally, we are gratefully indebted to various institutions for making IMFACO possible. To the Generalitat Valenciana, grants PROMETEO/2021/063 and CIAORG/2021/11, as well as to the Operations Research Center (Centro de Investigación OperativaCIO) and the Department of Statistics, Mathematics and Informatics of the Miguel

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Hernández University of Elche for financial support. And to the City of Elche for generously allowing us to use the Congress Center. Elche, Spain

José M. Amigó María J. Cánovas Marco A. López-Cerdá Manuel López-Pellicer

Contents

The Mathematical Research of Juan Carlos Ferrando . . . . . . . . . . . . . . . . . Manuel López-Pellicer

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A Class of Monothetic Reflexive Groups and the Weil Property . . . . . . . . L. Außenhofer

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Reciprocation and Pointwise Product in Vector Lattices of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gerald Beer and M. Isabel Garrido Lipschitzian Stability in Linear Semi-infinite Optimization . . . . . . . . . . . . M. J. Cánovas and J. Parra

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Bounded Duality in Topological Abelian Groups . . . . . . . . . . . . . . . . . . . . . . 113 M. J. Chasco and E. Martín-Peinador Topological Properties of the Weak and Weak∗ Topologies of Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Saak Gabriyelyan The Degree of Non-densifiability of Bounded Sets of a Banach Space and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 G. García and G. Mora A New Tour on the Subdifferential of the Supremum Function . . . . . . . . . 167 Abderrahim Hantoute and Marco A. López-Cerdá Optimality Conditions for Quasi Proper Solutions in Multiobjective Optimization with a Polyhedral Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 L. Huerga, B. Jiménez, and V. Novo

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On Distinguished Spaces C p (X) of Continuous Functions . . . . . . . . . . . . . 213 J. K¸akol and A. Leiderman Variational Convexity of Functions in Banach Spaces . . . . . . . . . . . . . . . . . 237 Pham Duy Khanh, Vu Vinh Huy Khoa, Boris S. Mordukhovich, and Vo Thanh Phat Commutators on Power Series Spaces Over Non-Archimedean Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 ´ Wiesław Sliwa and Agnieszka Ziemkowska-Siwek

The Mathematical Research of Juan Carlos Ferrando Manuel López-Pellicer

Dedicated to Juan Carlos Ferrando on the occasion of his 65th birthday.

Abstract This chapter is based on my invited talk at the International Meeting on Functional Analysis and Continuous Optimization dedicated to Juan Carlos Ferrando at the occasion of his 65 birthday at University Miguel Hernández in Elche, Spain, on June 16–17, 2022. We examine several topics of the research work of Professor Juan Carlos Ferrando. After the introductory section, this chapter is divided into seven sections, which include his research on Topological Vector Spaces, on Nikodým boundedness theorem, on distinguished Fréchet spaces, on C p -theory, on Ck -theory, on the weak topology of Ck (X ) and on the bidual of C p (X ). The section Research on Topological Vector Spaces contains four subsections. The proofs of Professor Ferrando are very clear and elegant. We have included several proofs, mainly in the sections devoted to C p -theory and Ck -theory, developed in the last five years. Keywords Closed graph theorem · C p and Ck theories · Distinguished fréchet spaces · Locally convex spaces · K-analytic spaces · Metrizability on precompacts · Nikodým boundedness properties · Normed spaces · Spaces of scalar-valued continuous functions · Spaces of vector-valued functions · Strongly barrelled conditions

M. López-Pellicer (B) IUMPA, Professor Emeritus at Universitat Politècnica de València, 46022 Valencia, Spain e-mail: [email protected] URL: https://rac.es/sobre-nosotros/miembros/academicos/numerarios/122/ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_1

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1 Introduction The first time I met Juan Carlos Ferrrando was in 1984 when I was the director of the Institute of Educational Sciences, where he visited me asking if I could be advisor of his Doctoral Thesis in Mathematics. I invited him to assist to the weekly seminar I had with my doctoral students Jesús Ferrer, who passed away in February 2022, José Ramón Ferrer, Valentín Gregori, José Más and Luis Manuel Sánchez Ruiz. Juan Carlos began to assist a couple of months later, after obtaining a chair in Mathematics at high school. Immediately, I was impressed by his speed solving problems and by his ability in the construction of examples and counterexamples. He obtained his Ph.D. degree in Mathematics in 1987 in the University of Valencia, Spain. The title of his Doctoral Thesis was Some classes of locally convex spaces related with closed graph theorem. Since then, we have been working together in more that thirty papers during 35 years. His mathematical genealogy from Gauss is depicted below, where each mathematician is the advisor of the one underneath. Von Staudt was not advisor from Felix Klein, but Klein is also a descendant from Gauss through C. L. Gerling (Göttingen 1812) and J. Plücker (Marburg 1823). Carl Friedich Gauss (Helmstedt, 1799) K. G. C. von Staudt (Erlangen-Nürnberg, 1822) Eduardo Torroja Caballé (Universidad Central, 1873) & Felix Klein (Universität Bonn, 1868) Julio Rey Pastor (Madrid and Göttingen, 1909) Ricardo San Juan (Universidad Central de Madrid, 1933) Manuel Valdivia (Universidad Complutense de Madrid, 1963) Manuel López-Pellicer (Universidad de Valencia, 1969) Juan Carlos Ferrando (Universidad de Valencia, 1987)

He currently has a position as a professor at the Miguel Hernandez University of Elche, Spain, where he is full professor since 2003. In December of 2022 MathSciNet collected 120 publications from professor Ferrando with 420 citations. Then, he had 80 publications in Functional Analysis, 32 in General Topology and 4 in Measure and Integration. In Scopus, he had 456 citations. Scopus does not count book citations, around 40 for the Ferrando coauthored book [76]. The pages 5 and 6 of their research papers in MathSciNet contain his initial research in locally convex spaces and part of his results obtained in this period are gathered in our coauthored book [76] with Luis Manuel Sánchez Ruiz. The monograph [94], Descriptive topology in selected topics of functional analysis, Developments in Mathematics 24, Springer (2011), contains 12 references of papers authored or coauthored by professor Ferrando. The book [9], Topological vector spaces and their applications, Springer Monographs in Mathematics, Springer, Cham (2017) have 7 references of Ferrando’s papers. Manuel Valdivia, Alexander Grothendieck and Michel Talagrand have 9, 6 and 6 references in this book, respectively. This shows clearly the relevance of Ferrando work since the beginning of his research in Functional Analysis.

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Next table contains Ferrando’s book [76] and his nine papers with more citations. Citations 35 31 22 17 15 13 13 11 11 11

Metrizable barrelled spaces, 1995 [76] Tightness and distinguished Fréchet spaces, 2006 [61] On precompact sets in spaces Ck (X ), 2013 [48] The dual of the locally convex space C p (X ), 2014 [64] Quasi-Suslin weak duals, 2008 [62] Characterizing P-spaces X in terms of C p (X ), 2015 [65] Metrizable bounded sets in C(X ) spaces and distinguished C p (X ) spaces, 2019 [50] On sequential barrelledness, 1991 [79] Necessary and sufficient conditions for precompact sets to be metrizable, 2006 [55] A note on spaces C p (X ) K -analytic-framed in R X , 2008 [47].

2 Research on Topological Vector Spaces 2.1 Strong Barrelledness Conditions Topological spaces and topological vector spaces in this chapter are supposed to be Hausdorff and locally convex, respectively, and we follow the usual notations and definitions of [32, 100, 102]. A locally convex (Hausdorff) space E is called barrelled if E verifies the Banach-Steinhaus boundedness theorem, i.e., every E pointwise bounded subset M of the dual E  of E is equicontinuous. Then E is barrelled if and only if each barrel of E (i. e., each absolutely convex, closed and absorbing set) is a neighborhood of the origin. A locally convex space E is called Baire-like if given an increasing sequence of closed absolutely convex subsets of E covering E, one of them is a neighborhood of the origin, [126]. The classic Amemiya-K¯omura theorem [1] guarantees that (i) each metrizable locally convex E is barrelled if and only if it is Baire-like, and (ii) if E is Baire-like and F is a dense barrelled subspace of E then F is Baire-like. A locally convex space E is called suprabarrelled in [138], or db in [128], if each increasing sequence of linear subspaces of E covering E has a dense barrelled member. This definition was generalized by transfinite induction by Rodríguez Salinas [123] as follows. If we call barrelled of class 0 to the barrelled spaces, for every successor ordinal α + 1 a locally convex space E is barrelled of class α + 1 if in each increasing sequence of linear subspaces of E covering E there is one of them which is dense and barrelled of class α, and for every limit ordinal α a locally convex space E is barrelled of class α if E is barrelled of class β for all β < α. A locally convex space E is totally barrelled (TB in brief) if given a sequence of linear subspaces of E covering E, one of them is Baire-like, [139]. A locally convex space E is unordered Baire-like (UBL for short) if each sequence of closed absolutely convex sets which covers E contains a neighborhood of the origin [134]. Full account of strong barrelledness conditions is given in [121, Chap. 9] and [105].

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Suprabarrelled spaces compose the class of barrelled spaces of class 1. Barrelled spaces of classes n and ω0 , the latter spaces called barrelled of class ℵ0 in [76], fit in the scheme of strong barrelledness properties as depicted in the following diagram Baire locally convex space ⇒ UBL ⇒ TB ⇒ barrelled of class ℵ0 ⇒ barrelled of class n + 1 ⇒ barrelled of class n ⇒ Baire-like ⇒ barrelled. Metrizable (L F) spaces are Baire-like but not suprabarrelled and each non normable Fréchet space contains a dense Baire-like subspace that is not suprabarrelled. Every infinite-dimensional Fréchet space contains a linear dense subspace which is totally barrelled but not unordered Baire-like, [128]. Examples of totally barrelled spaces that are not Baire can be found in [140]. In [128] Saxon and Narayanaswami proved that a metrizable barrelled space E is not suprabarrelled  of E such that if and only if there exists a linear subspace F of the completion E E ⊆ F and F is dominated by an (L F)-space, i. e., there is a stronger locally convex topology τ on F so that (F, τ ) is an (L F)-space. In [71] quasi-suprabarrelled spaces were introduced by removing the density requirement of the definition of suprabarrelled space. Quasi-suprabarrelled spaces have been called d spaces by Saxon in [127]. The following examples were obtained by Ferrando and coauthors in [71, 73], 1989 and 1992, respectively. Here and throughout the entire section ω = KN , being K the field of real or complex numbers. Example 1 In the space ω  ω N consider the sequence {E n : n ∈ N} of non× .n). . × ω × ϕω × ϕω × · · · , where ϕω means ϕ with barrelled subspaces E n = ω  the topology of ω. Then E = ∞ n=1 E n is a dense and barrelled subspace of the Fréchet space ω which is neither quasi-suprabarrelled nor dominated by any (L F)-space. Example 2 Equip Fn := ω n × 1 × 1 × · · · with the product topology τn , and define the (L F)-space (F, τ ) = lim (Fn , τn ). Then τ coincides with the relative →  topology of F = ∞ F as a linear subspace of ω and F is not suprabarrelled. n n=1 Assuming by induction that there exists a dense E in ω of class  barrelled subspace n G with G = ω × E × E × ··· s − 1 but not of class s, it turns out that G := ∞ n n n=1 is a dense barrelled subspace of ω N of class s but not of class s + 1. In both examples the use of the closed graph theorem for quasi-suprabarrelled or suprabarrelled spaces in the domain class is critical. Now, borrowing a classic result by Eidelheit (cf. [102, 31.4 (1)]) that states that each Fréchet space which is not Banach has a quotient isomorphic to ω, it follows that ([73], see also [76, Theorem 3.3.3]) Theorem 1 (Ferrando and López-Pellicer [73]) Given n ∈ N, each non-normable Fréchet space contains a dense barrelled subspace of class n − 1 but not of class n. As regards barrelled spaces of class ℵ0 , a detailed exposition is given in [76, Chap. 4]. Let us exhibit some separation examples. It was shown by Valdivia and Pérez Carreras in [139] that if E is a totally barrelled space which is not unordered Bairelike and F is a locally convex space, then the projective tensor product E ⊗π F is

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totally barrelled if and only if dim F < ℵ0 . Since E ⊗π F is barrelled of class ℵ0 whenever both E and F are barrelled of class ℵ0 and one of them is metrizable [76, Proposition 4.3.1], if E is an infinite-dimensional Fréchet space and F is a totally barrelled but not unordered Baire-like dense linear subspace of E (see [128]), it turns out that E ⊗π F is a dense barrelled subspace of class ℵ0 of E ⊗π E which is not π ω  ω contains totally barrelled. Particularly, if E = ω then the Fréchet space ω ⊗ a dense linear subspace which is barrelled of class ℵ0 but not totally barrelled. The next two examples can be found in [75]. Example 3 (Ferrando and López-Pellicer [75]) Each non normable Fréchet space of infinite dimension contains a dense barrelled subspace of class ℵ0 which is not totally barrelled. Example 4 (Ferrando and López-Pellicer [75]) If (Ω, Σ, μ) is a nontrivial measure space, L p (μ), with 1 ≤ p < ∞, has a dense subspace which is barrelled of class ℵ0 but not totally barrelled. If (Ω, Σ) is a measure space, it was established in [72] by Ferrando and LópezPellicer that the space ∞ 0 (Σ) of all scalarly-valued Σ-simple functions f : Ω → K equipped with the supremum-norm is barrelled of class ℵ0 . Since, according to a result of Arias de Reyna, if Σ is a non trivial σ-algebra the space ∞ 0 (Σ) is not totally barrelled (see [2]), it follows that ∞ 0 (Σ) is another example of a normed barrelled space of class ℵ0 which is not totally barrelled. Another class of strong barrelled spaces is that of baireled spaces, introduced  in [80]. A linear web of a locally convex space E is a countable family E n 1 ···n p :    of E suchthat E n 1 : n 1 ∈ N is an increasp, n 1 , . . . , n p ∈ N of linear subspaces  ing sequence covering E and if n 1 , . . . , n p−1 ∈ N p−1 then {E n 1 ···n p−1 n p : n p ∈ N} is increasing and verifies that ∞ n p =1 E n 1 ···n p−1 n p = E n 1 ···n p−1 . A baireled space is a locally convex space E such that each linear web in E contains a strand   E m 1 ···m p : p ∈ N of barrelled and dense spaces. Baireled spaces are strictly located between totally barrelled spaces and barrelled spaces of class ℵ0 , and baireledness is transmitted from dense subspaces and inherited by closed quotients, countable-codimensional subspaces and finite products. If E is baireled and metrizable and F is unordered Baire-like, then E ⊗π F is baireled [80, Proposition 4]. Hence if E is a metrizable totally barrelled space which is not unordered Baire-like, then E ⊗π 2 is baireled but not totally barrelled. Nonbaireled spaces which are barrelled of class ℵ0 are obtained as usual in each nonnormable Fréchet space by Eidelheit’s quotient theorem after showing that ω contains a dense subspace E of those characteristics. Main result of [109] reveals the Σ-simple scalarly-valued function space ∞ 0 (Σ) over a σ-algebra Σ of subsets of a given set Ω endowed with the supremum norm is baireled.

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2.2 Barrelled Spaces of Vector-Valued Functions Let (Ω, Σ) be a nontrivial measurable space and X be a normed space over K. If B(Σ, X ) denotes the normed space of X -valued functions defined on Ω that are the uniform limit of a sequence of Σ-simple X -valued functions defined on Ω endowed with the uniform convergence topology, the research on the barrelledness of locally convex spaces of vector-valued functions starts in 1982 when J. Mendoza shows that B(Σ, X ) is barrelled if and only if X is barrelled (cf. [23, 112]). If K is a compact space and C (K , X ) denotes the linear space of all X -valued continuous functions defined on K endowed with the compact-open topology, the following result, also due to Mendoza, characterizes the barrelledness of C (K , X ) in terms of X (see [113]). Theorem 2 (Mendoza [113]) C (K , X ) is barrelled if and only if both C (K ) and X are. If now Ω stands for a locally compact space and C0 (Ω, X ) denotes the space over K of continuous functions f : Ω → X vanishing at infinity (i. e., such that for > 0 there is a compact set K f, in Ω such that f (ω) < for ω ∈ Ω\K f, ) equipped with the supremum norm, the following result (cf. [53]) answers a question raised by J. Horváth. Theorem 3 (Ferrando et al. [53]) If Ω is a normal locally compact space, then C0 (Ω, X ) is barrelled if and only if X is barrelled. If c0 (Γ, X ) denotes the linear space of all X -valued functions defined on Ω such that for each > 0 the set {ω ∈ Ω : f (ω) > } is finite, provided with the supremum norm, using the fact that each compact subset of a discrete topological space (hence locally compact and normal) is finite, it holds that c0 (Γ, X ) = C0 (Γ, X ) whenever Γ is endowed with the discrete topology. So we have that c0 (Γ, X ) is barrelled if and only if X does. In [77] is shown that c0 (Γ, X ) is ultrabornological or unordered Baire-like if and only if X enjoys the corresponding property. This research was continued in [110], where it is proved that c0 (Γ, X ) is suprabarrelled if and only if X is suprabarrelled, in [111], where is shown that c0 (Γ, X ) is suprabarrelled of class p if and only if X barrelled of class p for every p ∈ N, in [99], where among others properties it is shown that c0 (Γ, X ) is totally barrelled if and only if X is totally barrelled and, finally, in [108], where it is shown that c0 (Γ, X ) is baireled if and only if X is baireled. In summary: Theorem 4 Let Ω be a nonempty set, X be a normed space and p ∈ N . Then c0 (Ω, X ) is barrelled of class p, baireled or totally barrelled if and only if X is, respectively, barrelled of class p, baireled or totally barrelled. As regards the spaces L p (μ, X ) the following results come from [25, 31]. Theorem 5 (Drewnowski et al. [31]) If (Ω, Σ, μ) is an atomless finite measure space and X is a normed space, then L p (μ, X ) is barrelled for 1 ≤ p < ∞.

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Theorem 6 (Díaz et al. [25]) If (Ω, Σ, μ) is an atomless finite measure space and X is a normed space, then L ∞ (μ, X ) is barrelled. In [43] we obtained the following generalization of the latter two theorems. Theorem 7 (Ferrando et al. [43]) If (Ω, Σ, μ) is an atomless finite measure space and X a normed space, then L p (μ, X ) is barrelled of class ℵ0 for 1 ≤ p ≤ ∞.

2.3 Metrizability of Precompact Sets A locally convex space E is called trans-separable if for every absolutely convex neighborhood of zero U in E there exists a countable subset NU of E such that E = NU + U . Clearly, a locally convex space E is trans-separable if and only if E is isomorphic to a subspace of a product of separable Banach spaces. Linear subspaces, locally convex products, completions, and linear continuous images of trans-separable locally convex spaces are trans-separable. convex     If E is a locally space with topological dual E  , then clearly E, σ E, E  and E  , σ E  , E are always trans-separable spaces. Here N is equipped with the discrete topology and NN with the product topology. A completely regular space X is quasi-Souslin (cf. [140]) if there is a map ϕ NN into the family of all (countably compact) subsets of X such that: (i)   from N ϕ (α) : α ∈ NN = X , and (ii) if a sequence {αn }∞ n=1 in N converges to α and ∞ xn ∈ ϕ (αn ) for all n ∈ N, then {xn }n=1 has an cluster point in X contained in ϕ (α). Since each metrizable quasi-Suslin locally convex space is separable, it turns out that each quasi-Suslin locally convex space is trans-separable. In paper [55] we get the following applicable result. Theorem 8 (Ferrando et al. [55]) In order for [pre] compact sets of a locally convex space E to be metrizable, it is both necessary and sufficient that E  endowed with the topology τc of uniform convergence on the compact sets of E [respectively, with the topology τ pc of uniform convergence on the precompact sets of E] be trans-separable. Since every quasi-Souslin locally convex space is trans-separable, our previous theorem includes that if (E  , τc ) is quasi-Souslin, then all compact sets in E are metrizable, a result obtained by Valdivia in [140, 1.4.3 (27)]. On the other hand, in [19] Cascales and Orihuela introduced a large class G of locally convex spaces that have a fundamental base of neighborhoods of 0 ordered by ω ω , including (L F)spaces and (D F)-spaces, and they proved that every precompact set of a locally convex space in class G is metrizable. Moreover in [20] they get the following interesting result: Theorem 9 A weakly compact set Y in a locally convex space E in class G is weakly metrizable if and only if Y is contained in a weakly separable set. The following three theorems from [62] sheds light on the preceding facts.

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     Theorem   10  If E ∈ G, both its weak* dual E , σ E , E and its Grothendieck dual E , τ pc , where τ pc is the topology of uniform convergence on the precompact sets in E, is a quasi-Suslin space. This property enables to extend to this wide class G of locally convex spaces several properties which were only known for particular classes of locally convex spaces. So, if M denotes the class of locally convex spaces having quasi-Suslin weak* duals, it follows from the previous theorem that G ⊆ M and that every precompact set of a space in class G is metrizable, as stated. Although class M is strictly wider than class G, there is one important case where both classes coincide. Recall that a locally convex space E is ∞ -barrelled if every weak* bounded sequence in E  is equicontinuous. Theorem 11 For an ∞ -barrelled space E, it happens that E ∈ G if and only if E ∈ M. Class M is the best known where the thesis of classic Kaplansky’s theorem holds. Let us recall that the tightness t (X ) of a topological space X is the smallest cardinal κ such that for every set A ⊆ X and each x ∈ A there exists a set B ⊆ A with |B| ≤ κ such that x ∈ B. In particular X has countable tightness if t (X ) ≤ ℵ0 . Theorem 12 Let E be a locally convex space. If E ∈ M, then E (weak) has countable tightness. We remark that Theorem 9 follows as a striking consequence from the below Theorem 13. It implies that if C p (X ) is a Lindelöf Σ-space and L ⊆ C p (X ) is separable there is a separable submetrizable Lindelöf Σ-space (Y, τ ) such that L is embedded into C p (Y, τ ). Moreover, from the Theorem 13 it can be readily shown that C p (X ) is analytic if and only if C p (X ) is separable and admits a compact resolution. Recall that a nonempty set Z is said to have a resolution if Z is covered by a family {Aα : α ∈ NN } of subsets such that Aα ⊂ Aβ for α ≤ β coordinatewise, i. e., such that α (i) ≤ β (i) for every i ∈ N. A compact resolution is a resolution in a topological space formed by compact subsets. The two times referred Theorem 13 is the main theorem of [60] and it use the cardinal functions density d (X ), weight w (X ), network weight nw (X ) and Nagami index N ag (X ). We appeal to Arkhangel’ski˘ı’s book [3] for the definition of those indices. Theorem 13 (Ferrando et al. [60]) If X is a topological space and L ⊆ C p (X ) there exists a space Y and two completely regular topologies τ  ≤ τ on Y such that 1. L is embedded in C p (Y, τ ), 2. N ag (Y, τ ) ≤ N ag (υ X ),

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  3. w Y, τ  ≤ d (L), 4. nw (Y, τ ) ≤ max {N ag (X ) , d (L)} and 5. d (L) ≤ max {N ag (L) , d ((Y, τ )}. Remark 1 In Sect. 6, after Theorem 45, we give the definition of trans-separability for a uniform space (X, N ) and in Theorem 46 we give the main result of the paper [56] that characterizes those uniform spaces which are trans-separable, by showing that a uniform space (X, N ) is trans-separable if and only if every pointwise bounded N -uniformly equicontinuous set of functions in C (X ) is metrizable in Ck (X, τN ).

2.4 Closed Graph Theorems There are a number of papers of professor Ferrando that contain a closed graph theorem. Here we shall exhibit three of them which are particularly useful. Recall that a locally convex space E is called quasi-suprabarrelled (cf. [71]) if given an increasing sequence of subspaces of E covering E, there is one of them which is barrelled. A locally convex space F is called a Γr -space (cf. [135, Theorem 2]) if every linear map T : E → F from a barrelled space E into F with closed graph is continuous. Each Br -complete space is a Γr -space, so every Fréchet space is a Γr -space. For a definition of Br -complete space and an account of classic closed graph theorems, see [103, Chap. 7]. The first closed graph theorem of our particular selection comes from [71]. Theorem 14 (Ferrando and López-Pellicer [71]) Assume that E is a quasisuprabarrelled space and let {Fn : n ∈ N} be an increasing sequence of linear subspaces of a locally convex space F covering F. Assume that each space Fn is dominated by a Γr -space. If T is a linear map from E into F with closed graph, then T is continuous. A topological space (X, τ ) is said to have a relatively countably compact resolution if X has a resolution consisting of relatively countably compact sets. Since Valdivia’s quasi-Suslin spaces have a relatively countably compact resolution, the following closed graph theorem (taken from [57]) extends Valdivia’s [140, I.4.2 (11)], and the case E = F (previously considered in [96] in the locally convex setting) extends a classic result of De Wilde and Sunyach that states that each Baire K -analytic locally convex space (a completely regular space X is K -analytic if it is ˇ the continuous image of a Cech-complete and Lindelöf space, see in Sect. 4.1 other definition) is a separable Fréchet space (see [140, I.4.3 (21)]). Theorem 15 (Ferrando et al. [57]) Let E and F be topological vector spaces such that E is Baire and F admits a relatively countable compact resolution. If T : E → F is a linear map with closed graph, then T is continuous. If E = F, then E is a separable Fréchet-space.

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Recall that a nonempty topological space (X, τ ) is called Fréchet-Urysohn if for every subset A of X and any point x ∈ A, where A denotes the closure of A in X , there exits a sequence of points of A converging to x. The following version of the closed graph theorem for topological groups can be found in [63] and it is connected with earlier results of Grothendieck, Pryce, Floret, Talagrand and others, concerning Eberlein-Šmulian type theorems and gave rise to the introduction of a large class of topological spaces that admit a relatively countable compact resolution with the following result connected with closed graph theorems obtained by Hoffman-Jørgensen, Martineau and Valdivia, among others. Theorem 16 (Ferrando et al. [63]) Let X and Y be topological groups such that X is Baire and Fréchet-Urysohn and Y admits a relatively countable compact resolution. If T : X → Y is a group homomorphism with closed graph, then T is continuous. Note that a Fréchet-Urysohn additive topological group G for which every null such that each subsequence {yn }∞ sequence {xn }∞ n=1 is a K -sequence (i. e.,  n=1 of ∞ ∞ {xn }n=1 has a subsequence {z n }n=1 so that ∞ z converges in G) is a Baire space n n=1 [12, Theorem 3].

3 Research on Nikodým Boundedness Theorem A subset B of an algebra A of subsets of a set Ω is a Nikodým set for the Banach space ba(A) of bounded finitely additive scalar measures defined on A, if every B-pointwise bounded subset M of ba(A) is a bounded subset of ba(A) endowed with the variation norm or, equivalently, M is uniformly bounded in A. For a σ-algebra Σ of subsets of a set Ω, we have that Σ is a Nikodým set for the Banach space ba(Σ). This property is the famous Nikodým-Grothendieck boundedness theorem, which is a good test for uniform boundedness in ba(Σ) with many applications in Banach spaces and in Measure theory. It was first obtained in 1933 by Nikodým for the σ-algebra of all subsets of N, in 1951 Dieudonné give the proof for the σ-algebra Σ of all subsets of a set Ω. The last version for a σ-algebra Σ of subsets of a set Ω is due to Grothendieck. A first deep improvement of this theorem was obtained by M. Valdivia in 1979 getting that if (Σn : n ∈ N) is an increasing covering of a σ-algebra Σ there exist a Σ p which is a Nikodým set for the Banach space ba(Σ) (see [137, 142]). A second improvement was obtained by Ferrando and López-Pellicer in the paper [72], where the authors show that for each web {Σσ : σ ∈ N p , p ∈ N} in a σ-algebra Σ of subsets of Ω and for each q there exists a finite chain (σn : σn ∈ Nn , 1 ≤ n ≤ q) such that each Σσn , 1 ≤ n ≤ q, is a Nikodým set for the Banach space ba(Σ), where, by chain definition, the first n components of σn+1 are the components of σn . After this result, Valdivia asked if for each web {Σσ : σ ∈ N p , p ∈ N} in a σalgebra Σ there exists a chain (σn : σn ∈ Nn , n ∈ N) such that each Σσn , n ∈ N, is a Nikodým set for ba(Σ). This problem was solved on [109] in 1997 by the baireledness of the space of Σ-simple functions provided with the supremum norm.

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Motivated by these results a subset B of an algebra A of subsets of a set Ω is named a strong Nikodým set for ba(A) if for each increasing covering (Bn : n ∈ N) of B there exists a Bm which is a Nikodým set for ba(A). Moreover, if for each web {Bσ : σ ∈ N p , p ∈ N} in B there exists a chain (σn : n ∈ N) such that each Bσn , n ∈ N, is a Nikodým set for the space ba(A) then B is called a web Nikodým set for the space ba(A). Problem 1 in [142] asks if the properties of Nikodým and strong Nikodým are equivalents or not in an algebra A of subsets of a set Ω. A partial solution to this 2013 Valdivia’s still open question is provided in the paper [69], where it is provided a class of rings of subsets of a set for which the equivalence between Nikodým, strong Nikodým and web Nikodým properties hold. The ring of subsets of density zero of N belongs to this class of rings, that, additionally, do not have the Grothendieck property. The name Grothendieck property is motivated by the well known fact that a Banach space E is a Grothendieck space if its dual and bidual, E  and E  , verify that for every sequence of E  the E-pointwise convergence to 0 implies its E  -pointwise convergence to 0. The current interest in Grothendieck spaces is motivated by several open problems. Analogously, it is said that an algebra A of subsets of a set Ω has the Grothendieck property if the completion of the linear hull of the characteristic functions of its elements, endowed with the supremum norm, is a Grothendieck space. This is equivalent to the property that the A-pointwise convergence to 0 of a bounded sequence of ba(A) implies its ba(A) -pointwise convergence to 0. This characterization motivates that a subset B of an algebra A of subsets of a set Ω is called a Grothendieck set for ba(A) if the B-pointwise convergence to 0 of a bounded sequence of ba(A) implies its ba(A) -pointwise convergence to 0. Each web in previous mentioned class of rings contains chains whose components are Nikodým sets that are not Grothendieck sets. This motivates to find algebras of subsets such that each web contains a chain of Grothendieck sets, considering also the Grothendieck sets version of the 2013 Valdivia open question on Nikodým sets. The first step solving this question was given in [70], where it was proved that if an algebra A of subsets of a set Ω has both the Nikodým and Grothendieck properties then in each increasing covering (An : n ∈ N) of A there exists A p which is a Grothendieck set for the Banach space ba(A). In particular, every increasing covering (Σn : n ∈ N) of a σ-algebra Σ contains a Σ p which is a Grothendieck set for ba(Σ). This result motivates that in [107] it is proved the following general and surprising property. Let B be a subset of an algebra A of subsets of a set Ω such that B and A are, respectively, a web Nikodým set and a Grothendieck set for the space ba(A), then for each web {Bσ : σ ∈ N p , p ∈ N} in B there exists a chain (σn : σn ∈ Nn , n ∈ N) such that each Bσn , n ∈ N, is both a Nikodým and a Grothendieck set for ba(A). Hence for each web {Σσ : σ ∈ N p , p ∈ N} in a σ-algebra Σ of subsets of Ω there exists a chain (σn : σn ∈ Nn , n ∈ N) such that each Σσn , n ∈ N, is both a Nikodým and a Grothendieck set for ba(Σ).

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4 Research on Distinguished Fréchet Spaces Following Dieudonné and [26] a Fréchet space E is called distinguished    Schwartz if its strong dual F = E  , β E  , E is barrelled, which always happens if E is a Banach if  Clearly, E is distinguished   and only if each bounded set in  space. E  , σ E  , E  is contained in the σ E  , E  -closure of a bounded subset of E, hence, roughly speaking, E is a large subspace of its weak* bidual E  .

4.1 Tightness and Distinguished Fréchet Spaces Let us recall again that the tightness t (X ) of a topological space X is the smallest cardinal κ such that for every set A ⊆ X and each x ∈ A there exists a set B ⊆ A with |B| ≤ κ such that x ∈ B. On the other hand, the character χ (E) of a locally convex space E is the smallest cardinal for a base of neighborhoods of the origin. In terms of these two indices, classic Kaplansky’s reads  as each locally convex space  theorem  E satisfies both t (E) ≤ χ (E) and t E, σ E, E  ≤ χ (E). Note   that aseparable locally convex space need not have countable tightness, since t RR = χ RR = c. It is well known that a completely regular space X is  K -analytic if thereexists  T (α) : α ∈ NN = X a map T : NN → 2 X with each T (α) compact such that ∞ N and if {αn }∞ n=1 converges to α in N and x n ∈ T (αn ) for every n ∈ N, then {x n }n=1 has a cluster point x ∈ T (α), i. e., if there map (an usc   is an upper semi-continuous T (α) : α ∈ NN = X . If T is countably map, in brief) T : NN → 2 X such that compactly-valued we recover the definition of quasi-Suslin space. Valdivia showed     [140, pp. 65–66] that if E is a Fréchet space, the bidual E  = E  , β E  , E of E equipped  topology is always quasi-Suslin, but is K -analytic if and  withthe weak* only if E  , μ E  , E  is barrelled, where μ (E, F) is the Mackey topology of the dual pair E, F. Problem 3 stated in [16] asks if a (D F)-space E with countable tightness must to be quasibarrelled, where quasibarrelled means that the bounded subsets of the strong dual of E are equicontinuous. The solution of this problem is provided in the paper [61] where it is proved that a (D F)-space, even more generally, a dual metric space, has countable tightness if and only if it is quasibarrelled. This result solves the mentioned [16, Problem 3] and, as a consequence, the authors get in [61, Corollary 4] the following is theorem: Theorem 17 (Ferrando et al. [61]) A Fréchet space E is distinguished if and only if its strong dual F has countable tightness, i.e., t (F) ≤ ℵ0 . This result provides a characterization of a complex vector topological property by tightness, a simple property related to cardinality. Its proof is obtained by new ideas relating vector topological properties with the bounding and dominating cardinals, named b and d, that enable the authors to prove that the tightness of the strong dual E  of the Grothendieck-Köthe nondistinguished Fréchet space E is d, and that  the tightness of E  endowed with the weak topology verifies b ≤ t E  , σ(E  , E  ≤ d.

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4.2 Examples and One Counterexample The first example of a nondistinguished Fréchet space by Grothen  was provided dieck and Köthe (cf. [102]). This is the echelon space λ, ν(λ, λ× ) of all numerical  double sequences x = (xi j ) such that i,∞j=1 |ai(n) j x i j | < ∞ for each n ∈ N. The steps (n) a(n) = (ai(n) ) are defined so that a = j for i ≤ n and all j ∈ N and ai(n) j ij j = 1 for i > n and all j ∈ N. Since λ is an echelon space, it is a perfect sequence space  × ×  × ) and λ, ν(λ, λ ) = λ . By and a Fréchet space in its normal topology ν(λ, λ  ×  ×  , β λ , λ of the Grothendieck-Köthe the previous theorem, the strong dual λ   Example 5] space λ, ν(λ, λ×) has uncountable  tightness. Since according   to [61,  it turns out that t λ× , σ λ× , λ > ℵ0 , it follows that λ , σ λ , λ× is not  K analytic [16, Theorem 4.6]. But according to Valdivia theorem λ , σ λ , λ is a quasi-Suslin space. So we get the following additional information about the space  λ, ν(λ, λ× ) in [61]. Example 5 (Ferrando et al. [61]) The weak* bidual of the Grothendieck-Köthe space is a quasi-Suslin locally convex space which is not K -analytic. The first nondistinguished Fréchet space whose weak bidual is quasi-Suslin but not K -analytic is due to M. Valdivia. In addition to the Example 5, professor Ferrando and his coauthors have provided a rich supply of (D F)-spaces whose weak duals are quasi-Suslin but not K -analytic, including the spaces Ck (κ), for κ any cardinal of uncountable cofinality c f (κ). In [61, Corollary 4] we find the following example. Example 6 (Ferrando et al. [61]) If c f (κ) > ℵ0 then Ck ([0, κ)), where here κ is regarded as a (limit) ordinal, is quasi-Suslin but not K -analytic. Lets finally mention a counterexample taken from [63]. It exhibits a countably compact topological space G whose product G × G cannot be covered by an ordered family {Aα : α ∈ NN } of relatively countably compact sets. This shows that quasiSuslin spaces are not productive (see also [82]). Let X be a discrete space of cardinality c and let X 1 and X 2 be two subspaces of X such that (i) X 1 ∩ X 2 = ∅, (ii) X 1 ∪ X 2 = X , and (iii) |X 1 | = |X 2 | = c. By ˇ extension σ β (iii) there exists a bijection σ from X 1 onto X 2 whose Stone-Cech is a homeomorphism from β X 1 onto β X 2 . Since X is a discrete space, we have βX βX βX βX βX X 1 ∩ X 2 = ∅ and X 1 ∪ X 2 = X . If Y is a subspace of X , then we can βX identify βY with Y . Hence β X 1 ∩ β X 2 = ∅ and β X 1 ∪ β X 2 = β X . Moreover, if βX N is a countable infinite subspace of X then |N | = |β N | = |βN| = 2c . Now define a homeomorphism ϕ : β X → β X by ϕ (x) = σ β (x) if x ∈ β X 1 and ϕ (x) = (σ β )−1 (x) if x ∈ β X 2 . Clearly ϕ (ϕ ( p)) = p for every p ∈ β X , and p ∈ X if and only if ϕ ( p) ∈ X . Since ϕ(β X 1 ) = β X 2 and ϕ(β X 2 ) = β X 1 , the map ϕ does not have fixed points. If N denotes the family of all countable infinite subsets of X , put  βX Z := {N : N ∈ N } and denote by M the family of all countable infinite subsets of Z . Since |N | = cℵ0 then |Z | = cℵ0 × 2c = 2c and hence |M| = 2c . So, if m is

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the first ordinal of cardinality 2c , one gets that M = {Mα : 0 ≤ α < m}. Note that α < m implies that |α| = |[0, α)| < 2c and X is contained in Z . Moreover, it can be βX easily seen that if M ∈ M then |M | = 2c . Now it is possible to define inductively a       βX set Γ = yγ : 0 ≤ γ < m such that yα ∈ Mα \(Mα ∪ ϕ yγ : 0 ≤ γ < α ) for every 0 ≤ α < m. Example 7 (Ferrando et al. [63]) Setting G := X ∪ Γ , due to every countable infinite subset A of G is equal to Mα for some 0 ≤ α < m, it turns out that G contains a limit point of A. Therefore G is countably compact. On the other hand, the graph {( p, ϕ ( p)) : p ∈ β X } of the continuous map ϕ : β X → β X is closed in β X × β X , so that S := {(x, ϕ (x)) : x ∈ X } is a closed subspace of G × G homeomorphic to X . So S is uncountable and discrete, which prevents G × G to be covered by an ordered family {Aα : α ∈ NN } of relatively countably compact sets. It is well-known that any countable product of K -analytic (analytic) spaces is K -analytic (analytic). However, Example 7 applies to deduce that the quasi-Suslin property is not productive.

5 Research on C p -Theory Unless otherwise stated, X will stand for an infinite Tychonoff space. We denote by C p (X ) the linear space C(X ) of real-valued continuous functions on X equipped with the pointwise topology τ p . The topological dual of C p (X ) is denoted by L(X ), or by L p (X ) when provided with the weak* topology. We denote by Ck (X ) the space C(X ) equipped with the compact-open topology τk . Recall that X is a Lindelöf Σ-space if it is a continuous image of a space that can be perfectly mapped onto a second countable space [3, 117]. Also, X is a Lindelöf Σ-space if and only if is countably K -determined [124], i. e., if there is an upper semi-continuous (usc) map T  from a subspace Σ of NN into the family K (X ) of compact subsets of X such that {T (α) : α ∈ Σ} = X . This is equivalent to saying that (i) {T (α) : α ∈ Σ} covers X and (ii) if αn → α in Σ and xn ∈ T (αn ) for every n ∈ N the sequence {xn }∞ n=1 has a cluster point in T (α). Remind (see Sect. 4.1) that a space X is K -analytic (resp. quasi-Suslin) if there is a map T from NN into K (X ) (resp. into the family of countably compact sets in X ) such that (i) {T (α) : α ∈ NN } covers X and (ii) if αn → α in NN and xn ∈ T (αn ) for each n ∈ N the sequence {xn } has a cluster point contained in T (α) (see [140, I.4.2 and I.4.3]). Each σ-compact (σcountably compact) space is K -analytic (resp. quasi-Suslin). A space X is analytic if it is a continuous image of NN . Each analytic space is K -analytic, so quasi-Suslin and Lindelöf Σ, and each Lindelöf Σ-space is Lindelöf. A family N of subsets of a X is a network for X if for any x ∈ X and any open set U in X with x ∈ U there is some P ∈ N such that x ∈ P ⊆ U . The network weight nw (X ) of X is the least cardinality of a network of X , and a space X is called cosmic if nw (X ) = ℵ0 . Alternatively, X is a cosmic space if and only if it is a continuous image of a separable

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metric space [115]. So, each analytic space is cosmic. Conversely, every K -analytic cosmic space is analytic [94, Proposition 6.4]. Moreover, C p (X ) is a cosmic space if and only if X is cosmic [115, Proposition 10.5]. If X is a topological space, a resolution for X consisting of topologically bounded or compact sets is respectively called a topologically bounded or a compact resolution. If E is a locally convex space, a resolution for E consisting of bounded sets is called a bounded resolution. A family of bounded subsets of a locally convex space E that swallows the bounded sets is sometimes called a fundamental family of bounded sets. So, a fundamental bounded resolution for E is a bounded resolution that swallows the bounded subsets of E. According to [94, Theorem 3.2] or [129] each K -analytic space has a compact resolution. The converse holds true for a C p -space. Theorem 18 (Tkachuk [130, 2.8 Theorem]) C p (X ) has a compact resolution if and only if it is K -analytic. A space X is angelic if relatively countably compact sets in X are relatively compact and for every relatively compact subset A of X each point of A is the limit of a sequence of A, [84]. If C p (X ) is K -analytic, then υ X is a Lindelöf Σ-space by [118, Theorem 3.5]. So C p (υ X ) is angelic by Orihuela’s angelicity theorem [120, Theorem 3]. Hence C p (X ) is angelic as well. So the following property holds. Theorem 19 (Ferrando [36, Theorem 8]) The following are equivalent 1. C p (X ) is a quasi-Souslin space. 2. C p (X ) is K -analytic. Remind that a family N of subsets of a space X is a network modulo a family A of subsets of X if for each open set V of X and for every A ∈ A with A ⊆ V there is N ∈ N with A ⊆ N ⊆ V . So, a space is Lindelöf Σ if and only if it admits a countable network modulo a covering by compact sets [94, Proposition 3.5]. Hence, every cosmic space is a Lindelöf Σ-space. Let Σ ⊆ NN and A = {Aα : α ∈ Σ} be a family of subsets of X . For each (α, n) ∈ Σ × N let us define A (α | n) =



 Aβ : β ∈ Σ, β (i) = α (i) , 1 ≤ i ≤ n .

Clearly Aα ⊆ A (α | n) for every n ∈ N and A (α | n + 1) ⊆ A (α | n) for every (α, n) ∈ Σ × N. Since A (α | n) = A (β | n) whenever α (i) = β (i) for 1 ≤ i ≤ n, it turns out that E = {A (α | n) : α ∈ Σ, n ∈ N} is a countable family of subsets of X . The family E is called the envelope of A. The envelope of a family {Aα : α ∈ Σ} of subsets of a topological vector space E with Σ ⊆ NN is called limited if for each α ∈ Σ and each balanced neighborhood of the origin U in E there is n ∈ N with A (α | n) ⊆ nU . If {Aα : α ∈ Σ} is a family of subsets of E with limited envelope, each Aα is bounded in E. Lemma 1 (Ferrando [35, Lemma 2]) Let E be a topological vector space with dual E  . If E admits a Σ-covering {Aα : α ∈ Σ} with Σ ⊆ NN of limited envelope, there exists a Lindelöf Σ-subspace Z of R E such that E  ⊆ Z ⊆ R E .

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Proof We will say that a real-valued function f defined on E is E-bounded if for each x ∈ E there exist α ∈ Σ and n ∈ N such that x ∈ A (α | n) and f (A (α | n)) is a bounded subset of R. By Z we will denote the subset of R E consisting of all E-bounded functions on E. Let us prove that E  ⊆ Z . Indeed, fix u ∈ E  and x ∈ E, and since {Aα : α ∈ Σ} covers E, choose α ∈ Σ with x ∈ Aα . If we set U = u −1 {(−1, 1)} then U is a balanced neighborhood of the origin in E, hence there is n ∈ N such that x ∈ Aα ⊆ A (α | n) ⊆ nU . This implies that |u(y)| ≤ n for each y ∈ A (α | n). Thus u belongs to Z . E Now we prove that Z is a Lindelöf Σ-space. Let H be the closure of Z in R , where R designs the usual two points compactification of R. For each α ∈ Σ and n, p ∈ N set L α,n, p = { f ∈ H : f (A (α | n)) ⊆ [− p, p] }. The sets L α,n, p are compact since they are closed in H , and compose a countably family. Choose f ∈ Z and g ∈ H \Z . Since g ∈ H \Z , there exists y ∈ E such that g (A (α | n)) is unbounded in R for each (α, n) ∈ Σ × N for which y ∈ A (α | n). Since f ∈ Z there are α ∈ Σ and n ∈ N such that y ∈ Aα ⊆ A (α | n) and f (A (α | n)) is bounded in R, so there is p ∈ N such that f ∈ L α,n, p . But on the other hand g ∈ / L α,n, p since g is unbounded on A (α | n). Given that H is a compactification of Z , [3, Proposition IV.9.2] applies to conclude that Z is a Lindelöf Σ-space. Theorem 20 (Ferrando [35, Theorem 3]) υ X is a Lindelöf Σ-space if and only if C p (X ) admits a Σ-covering with limited envelope. Proof If υ X is a Lindelöf Σ-space, by a theorem due to Uspenski˘ı [3, Proposition IV.9.3] there is a Lindelöf Σ-subspace S of Rυ X such that C p (υ X ) ⊆ S. So, if K (S) denotes the family of all compact subsets of S, there is a subspace Σ of NN and a  usc map T : Σ → K (S) such that {T (α) : α ∈ Σ} = S. Since each T (α) is a that Bα = T (α) ∩ C p (υ X ) is a compact subset of S, and hence of Rυ X , it follows  bounded subset of C p (υ X ) for all α ∈ Σ. Clearly {Bα : α ∈ Σ} = C p (υ X ). If Φ : C p (υ X ) → C p (X ) denotes the restriction map defined by Φ ( f ) = f | X , then Φ is a continuous linear map from C p (υ X ) onto C p (X ). Thus if we set Aα = Φ (Bα ) for every α ∈ Σ we can see that {Aα : α ∈ Σ} is a covering of C p (X ) by bounded sets.   Letting as usual A (α | n) = Aβ : β ∈ Σ, β (i) = α (i) , 1 ≤ i ≤ n , we claim that the envelope {A (α | n) : (α, n) ∈ Σ × N} of {Aα : α ∈ Σ} is limited. Indeed, assume on the contrary that there are α ∈ Σ and a balanced neighborhood of the origin U in C p (X ) such that A (α | n)  nU for all n ∈ N. Then choose f n ∈ A (α | n) \nU   for each n ∈ N and a sequence {βn } ⊆ Σ with f n ∈ Φ Bβn and βn (i) = α (i) for 1 ≤ i ≤ n. Let {gn } ⊆ C (υ X ) with gn ∈ T (βn ) such that Φ (gn ) = f n for each n ∈ N.

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Since T is a usc map and βn → α, it follows that all subsequences of {gn } have a cluster point belonging to the compact subset T (α) of C p (υ X ). This fact ensures that {gn : n ∈ N} is a relatively countably compact subspace of C p (υ X ), hence a bounded subset of C p (υ X ). Consequently { f n : n ∈ N} must be a bounded subset of C p (X ). Since U is a balanced neighborhood of the origin in C p (X ) there is m ∈ N such that { f n : n ∈ N} ⊆ mU , which yields the contradiction f m ∈ mU . Conversely, if C p (X ) has a Σ-covering with limited envelope an application of Lemma 1 with E = C p (X ) produces a Lindelöf Σ-space Z such that L p (X ) ⊆ Z ⊆ RC(X ) . Since X is embedded in L p (X ), it follows that X ⊆ Z ⊆ RC(X ) . Using the fact that the realcompactification υ X of X coincides with the closure of X in RC(X ) , we deduce that the closure Y of X in Z is contained in υ X , that is, X ⊆ Y ⊆ υ X . But this implies that υ X = υY = Y (see [89, Theorem 8.6]). Since Y is a Lindelöf Σ-space because it is a closed subspace of the Lindelöf Σ-space Z , υ X is a Lindelöf Σ-space as stated. A space C p (X ) is said to be Lindelöf Σ-framed (or K -analytic-framed) in R X if there is a Lindelöf Σ-space (resp. a K -analytic space) S in R X such that C (X ) ⊆ S. Lemma 2 If C p (X ) is Lindelöf Σ-framed in R X , then υ X is a Lindelöf Σ-space and C p (X ) is angelic. Proof First statement after the conditional comes from Theorem 20. For the second use the first and [120, Theorem 3], since C p (X ) is angelic whenever C p (υ X ) is angelic. Lemma 3 (Ferrando-K¸akol [47, Lemma 1]) Let X be nonempty and Z be a subspace of R X . If Z has a countable network modulo a cover B of Z by pointwise bounded  subsets, then Y = {B : B ∈ B}, closures in R X , is a Lindelöf Σ-space such that Z ⊆ Y ⊆ RX . Proof Let N = {Tn : n ∈ N} be a countable network modulo a cover B of Z consisting of pointwise bounded sets. Set N1 = {T n : n ∈ N}, B1 = {B : B ∈ B}, closures in R X , and Y = ∪B1 . Let us show that N1 is a network in Y modulo the compact cover B1 of Y . In fact, if U is a neighborhood in R X of B, use B compactness to get a closed neighborhood V of B in R X contained in U . Since N is a network modulo B in Z there is n ∈ N with B ⊆ Tn ⊆ V ∩ Z , which implies that B ⊆ T n ⊆ U . According to Nagami’s criterion [3, IV.9.1 Proposition], Y is a Lindelöf Σ-space such that Z ⊆ Y ⊆ R X . Theorem 21 (Ferrando-K¸akol [47, Proposition 1]) The following asserts are equivalent 1. C p (X ) admits a bounded resolution. 2. C p (X ) is K -analytic-framed in R X . Proof Let {Aα : α ∈ NN } be a resolution for  C p (X ) of bounded sets, denote by Bα the closure of Aα in R X and put Z = {Bα : α ∈ NN }. Clearly each Bα is a

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compact subset of R X and Z is a quasi-Suslin space [15, Proposition 1] such that C p (X ) ⊆ Z ⊆ R X . As each quasi-Suslin space Z has a countable network modulo a resolution B of Z consisting of countably compact sets (see [34, Proof Theorem 8]) and every countable compact subset of R X is pointwise bounded, Lemma 3  assures that Y = {B : B ∈ B} is a Lindelöf Σ-space, hence Lindelöf, such that Z ⊆ Y ⊆ R X . As each set B with B ∈ B is compact, and {B : B ∈ B} is a resolution for Y , again Y is a quasi-Suslin space. Since every Lindelöf quasi-Suslin space is K -analytic and C p (X ) ⊆ Y ⊆ R X , it turns out that C p (X ) is K -analytic-framed in R X . For the converse, note that each K -analytic space has a resolution consisting of compact sets [129]. A space X is projectively σ-compact if each separable metrizable space Y that is a continuous image of X is σ-compact. Clearly, every σ-topologically bounded space is projectively σ-compact [5, Proposition 1.1], and every projectively σ-compact cosmic space is σ-compact (see [94, Proposition 9.4] or [119]). Theorem 22 (Arkhangel’ski˘ı-Calbrix [6, Theorem 2.3]) If C p (X ) is K -analyticframed in R X , then X is projectively σ-compact. Corollary 1 If C p (X ) admits bounded resolution, then X is projectively σ-compact. Proof This is a consequence of Theorems 21 and 22. The converse of the previous corollary does not hold, as the following example shows. Example 8 (Ferrando [67, Example 10]) There exists a projectively σ-compact space X such that C p (X ) does not admit a bounded resolution. Proof Let X be the one-point Lindelöfication of the ℵ1 -discrete space. Since X is a Lindelöf P-space, X is an ω-space, i. e., every continuous metrizable separable image of X is countable. Consequently, X is projectively σ-compact. Since X is a P-space, C p (X ) is pseudocomplete in the sense of Oxtoby, hence a Baire locally convex space. Assume C p (X ) admits a bounded resolution. By [96, Corollary 1] the space C p (X ) is metrizable, so X must be countable, a contradiction. Let us point out that, in addition, it can be easily seen that C p (X ) itself is not projectively σ-compact either. Theorem 23 (Ferrando-K¸akol [47, Corollary 1]) Let X be a cosmic space. C p (X ) has a bounded resolution if and only if X is σ-compact. Proof The ‘only if’ statement is consequence of Corollary 1 and the fact, mentioned earlier, that each projectively σ-compact cosmic space is σ-compact. For the ‘if’ part  K with each K n compact, the family {Aα : α ∈ NN } with note that if X = ∞ n=1 n   Aα = f ∈ C (X ) : supx∈K n | f (x)| ≤ α (n) , n ∈ N is a resolution for C (X ) consisting of pointwise bounded sets.

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Theorem 24 (Calbrix [13, Theorem 2.3.1]) If C p (X ) is analytic, then X is σcompact. Proof If C p (X ) is analytic, it is cosmic. Hence X is also a cosmic space [115, Proposition 10.5]. Since C p (X ) is K -analytic, it has a resolution of pointwise bounded sets (actually, of compact sets [129]). So, Theorem 23 ensures that X is σ-compact. Corollary 2 If X is metrizable, the following are equivalent. 1. C p (X ) is analytic. 2. X is σ-compact. 3. C p (X ) has a bounded resolution. Proof 1 ⇒ 2 follows from Theorem 24 and, as mentioned above, 2 ⇒ 3 always holds true. On the other hand, if C p (X ) has a resolution of pointwise bounded sets, then C p (X ) is K -analytic-framed in R X by Theorem 21 and angelic by Lemma 2. But if X is metrizable, C p (X ) is angelic if and only if X is separable [94, Corollary 6.10]. Consequently, for metrizable X , the fact that C p (X ) has a resolution of pointwise bounded sets entails that X is a cosmic space. So, Theorem 23 yields the implication 3 ⇒ 2. Finally, if X is a metrizable σ-compact space then X is separable. Thus C p (X ) is analytic by a classic result of Christensen [24, Theorem 3.7] (cf. Theorem 45 below). Hence 2 ⇒ 1. If E is a locally convex space, a set A in E is called (relatively) sequentially complete [65] if every Cauchy sequence {xn }∞ n=1 in E contained in A converges in E to a point x ∈ A (a point x ∈ E). Theorem 25 (Ferrando-K¸akol-Saxon [65, Theorem 3.1]) C p (X ) is covered by a sequence of relatively sequentially complete sets if and only if X is a P-space.  Proof Assume that C p (X ) = ∞ n=1 Q n , with Q n relatively sequentially complete be a uniformly bounded pointwise eventually confor every n ∈ N, and let { f n }∞ n=1 stant sequence in C p (X ) with limit f in R X . Let us denote by C b (X ) the Banach space of all continuous and bounded functions on X equipped with the supremum norm · ∞ . Fix k > 0 such that supn∈N f n ∞ ≤ k. Since C b (X ) ∩ Q n : n ∈ N is a countable covering of C b (X ), according to the Baire category theorem there is p ∈ N such that the closure B p of C b (X ) ∩ Q p in C b (X ) has an interior point in the norm topology. So, if D denotes the closed unit ball of C b (X ), there are  > 0 and h ∈ Q p with h + D ⊆ B p . Since f n ∈ k D for each n ∈ N, we have h + k −1 f n : n ∈ N ⊆ B p . As C b (X ) ∩ Q p is norm dense in B p , for each n ∈ N there is gn ∈ C b (X ) ∩ Q p with





gn (x) − h + k −1 f n (x)
0 with supg∈Q n |g (xn )| < αn . But [94, / Q n for every Lemma 9.5] provides f ∈ C (X ) with f (xn ) = αn , i. e., such that f ∈ n ∈ N, a contradiction. Thus X must be finite. A resolution for X is said to swallow a family N of sets in X if for each P ∈ N there is A ∈ A such that P ⊆ A. Theorem 27 (Tkachuk [130, 3.7 Theorem]) C p (X ) has a compact resolution that swallows the compact sets if and only if X is countable and discrete. A family N of subsets of a topological space X is called a cs ∗ -network at a point x ∈ X if for each sequence {xn }∞ n=1 in X converging to x and for each neighborhood Ox of x there is a set N ∈ N such that x ∈ N ⊆ Ox and the set {n ∈ N : xn ∈ N } is infinite [87]; N is a cs ∗ -network in X if N is a cs ∗ -network at each point x ∈ X . Theorem 28 (Ferrando-Gabriyelyan-K¸akol [46, Theorem 3.3]) C p (X ) has a resolution of bounded sets that swallows the bounded sets if and only if X is countable. In other words, C p (X ) has a fundamental resolution of pointwise bounded sets if and only if X is countable. Proof (Sketch) If C p (X ) admits a fundamental resolution of pointwise bounded sets one can fix [46, Theorem 3.3] a countable family of closed sets (some of them may be empty) K = {K n (α) : n ∈ N, α ∈ NN } in X enjoying the properties: 1. 2. 3. 4.

K n (α) ⊆ K n+1 (α) for every n ∈ N and each α ∈ NN . K n (α) ⊇ K n (β) for every n ∈ N whenever α ≤ β.  N n∈N K n (α) = X for each α ∈ N . For every increasing closed covering {Vn : n ∈ N} of X there exists γ ∈ NN such that K n (γ) ⊆ Vn for all n ∈ N. Then it turns out that the family N := {Nmn (α) : m, n ∈ N, α ∈ NN }, where  Nmn (α) :=

f ∈ C(X ) : | f (x)| ≤

1 ∀x ∈ K n (α) m



and Nmn (α) := {0} if K n (α) is empty, is a countable cs ∗ -network at the origin in C p (X ) (see [46, Proposition 3.2] or [41, Claim 108] for details). So, according to [125, Theorem 2.3], X must be countable.

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Recall that a locally convex space E is a quasi-(L B)-space if E has a resolution consisting of Banach disks, i. e., of absolutely convex bounded sets D whose linear span E D is a Banach space when equipped with the Minkowski functional of D as a norm. Theorem 29 (Valdivia [141]) If E is a quasi-(L B)-space, there exists a resolution for E consisting of Banach disks that swallows the Banach disks of E. Theorem 30 (Ferrando-Gabriyelyan-K¸akol [46, Proposition 3.6]) Let X be a Pspace. C p (X ) has a bounded resolution if and only if X is countable and discrete. Proof If X is a P-space then C p (X ) is locally complete [65, Theorem 1.1], i. e., each bounded set is contained in a Banach disk. So, according to Theorem 29 there exists a resolution for C p (X ) consisting of Banach disks that swallows the bounded sets in C p (X ). Therefore, X is countable by Theorem 28, and each countable P-space is discrete. Alternatively, one may use the facts that C p (X ) is a Baire space (note that C p (X ) is pseudocomplete [131, Sect. 1.5, p. 46] whenever X is a P-space and use [131, Problem 464]) and that each locally convex Baire space with a resolution of bounded sets is metrizable (see [96, Corollary 1]). Hence X must be countable, so discrete. Recall that a sequence {xn }∞ n=1 in a locally convex space E is called local null or Mackey convergent to zero [102, 28.3] if there is a closed disk B in E such that xn → 0 in the normed space E B . Each local null sequence in E is a null sequence. Theorem 31 (Ferrando [42, Theorem 12]) C p (X ) admits a resolution consisting of convex compact sets that swallows the local null sequences in C p (X ) if and only if X is countable and discrete. Proof We may assume that C p (X ) admits a resolution {Aα : α ∈ NN } of absolutely convex compact sets swallowing the local null sequences in C p (X ). If T : C p (υ X ) → C p (X ) denotes the restriction map T g = g| X we proceed as in [94, Proposition 9.14] to show that the family A = {T −1 (Aα ) : α ∈ NN } is a resolution for C p (υ X ) consisting of (absolutely convex) compact sets, with the additional benefit that A swallows the local null sequences in C p (υ X ). So, we may assume without loss of generality that X is realcompact or, equivalently, that C p (X ) is bornological [11]. Hence, we denote as above by {Aα : α ∈ NN } a resolution for C p (X ), with X realcompact, consisting of absolutely convex compact sets that swallows the local null sequences in C p (X ). Let M denote the family of all local null sequences in C p (X ). Since {Aα : α ∈ NN } swallows the members of M, the Mackey* topology μ (L (X ) , C (X )) of L (X ) is stronger than the topology τc0 on L (X ) of the uniform convergence on the local null sequences of C p (X ). As in addition σ (L (X ) , C (X )) ≤ τc0 , we conclude that   L (X ) , τc0  = C (X ).Moreover, since we are assuming that C p (X ) is bornological, its τc0 -dual L (X ) , τc0 is complete by [102, 28.5.(1)]. We claim that every compact set in X is finite. Indeed, if K is a compact set in X , the homeomorphic copy δ (K )

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of K in L p (X ) is compact,  i. e., δ (K) is a σ (L (X ) , C (X ))-compact set in L (X ). So, the completeness of L (X ) , τc0 , together with Krein’s theorem and the fact that τc0 is a locally convex topology of the dual pair L (X ) , C (X ), ensures that the weak* closure Q = abx (δ (K )) in L (X ), where abx (δ (K )) stands for the absolutely convex hull of δ (K ), is a compact set in L p (X ), hence a strongly bounded set. Since C p (X ) is quasi-barrelled [93, 11.7.3 Corollary], the strongly bounded sets in L (X ) are finite-dimensional. Therefore the set δ (K ), as a linearly independent system of vectors in L (X ), must be finite. Thus K is finite as well. Since υ X = X is a Lindelöf Σ-space by Lemma 2 and as we know each Lindelöf Σ-space with finite compact sets is countable [3, IV.6.15 Proposition], X is countable. So C p (X ) is a metrizable space. But in a metrizable locally convex space, the local null sequences and the null sequences are the same [102, 28.3.(1) c)]. Furthermore, if M is a compact set in the metrizable space C p (X ), then M lies in the closed absolutely convex cover of a null sequence { f n }∞ n=1 , [102, 21.10.(3)]. ⊆ A , thanks to the fact that Aγ is a closed absolutely convex So, if { f n }∞ γ n=1 set, it turns out that M ⊆ Aγ . Therefore {Aα : α ∈ NN } is a compact resolution for C p (X ) that swallows the compact sets of C p (X ). So, C p (X ) is a Polish space by Christensen’s theorem [41, Theorem 94]. But then [3, I.3.3 Corollary] asserts that X is discrete. The converse is obvious. Theorem 32 (Ferrando [42, Theorem 16]) C p (X ) has a resolution consisting of absolutely convex pointwise bounded sequentially complete sets that swallows the null sequences if and only if X is countable and discrete. Proof It can be readily seen that there is no loss of generality if we assume X to be realcompact. If C p (X ) has a resolution {Aα : α ∈ NN } of the stated characteristics a null sequence in C p (X ), there is γ ∈ NN such that f n ∈ Aγ for every and { f n }∞ n=1 is n ξi f i ∈ Aγ for every ξ ∈ 1 with ξ 1 ≤ 1 and Aγ is sequentially n ∈ N. Since i=1 ∞ ξi f i ∈ Aγ for every ξ ∈ 1 with ξ 1 ≤ 1. So, the complete, it follows that i=1 Banach disk 

∞  ξi f i : ξ ∈ 1 , ξ 1 ≤ 1 Q := i=1

is contained in Aγ . Now, it can be proved as in [102, 20.10.(6)] that Q = { f n : n ∈ N}00 , the absolute bipolar of the null sequence { f n : n ∈ N}. Since each local null sequence is a null sequence, the dual of L (X ) , τc0 is C (X ), so σ (L (X ) , C (X )) ≤ τc0 ≤ μ (L (X ) , C (X )). As C p (X ) is bornological, the space L (X ) is μ (L (X ) , C (X ))-complete. So, proceeding as in the proof of Theorem 31, with the help of Krein’s theorem we establish that each compact set in X is finite. Now, using the fact that the resolution {Aα : α ∈ NN } consists of pointwise bounded sets, Lemma 2 asserts that X is a Lindelöf Σ-space. Thus X must be countable, [3, IV.6.15 Proposition], so C p (X ) is metrizable. If M is a compact set in the metrizable space C p (X ), as mentioned above M lies in the closed absolutely convex cover of a null sequence { f n }∞ n=1 . So, if { f n } ⊆ Aγ then M ⊆ Aγ . Thus {Aα : α ∈ NN } is a resolution for C p (X ) that swallows the

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compact sets of C p (X ). Since each set Aα is precompact in C p (X ) and sequentially complete, the metrizability of C p (X ) ensures that Aα is compact in C p (X ). Hence C p (X ) is a Polish space by Christensen’s theorem. Thus X is discrete. The converse is clear, since each (absolutely convex) compact set in RN is pointwise bounded and sequentially complete. Theorem 33 (Ferrando [42, Theorem 33]) Let X be first countable. C p (X ) has a resolution of bounded sets that swallows the Cauchy sequences if and only if X is countable. Next we show that if X is a Lindelöf Σ-space and A is a (relatively) countably compact subset of C p (X ), then A is (relatively) compact. This guarantees in particular that C p (X ) is angelic. In fact, as every compact set in C p (X ) is Fréchet-Urysohn [39, Lemma 2], if A is relatively compact in C p (X ) and f ∈ A, there is a sequence { f n }∞ n=1 contained in A such that f n → f in C p (X ). So C p (X ) is angelic. Theorem 34 (Ferrando and López-Alfonso [68, Theorem 2]) If X is a Lindelöf Σspace, each functionally bounded (relatively) sequentially complete set in C p (X ) is (relatively) compact. Proof We consider the relatively sequentially complete setting, the other case is similar. By hypothesis there are a subset Σ of NN and a map T from Σ into the family K (X ) of compact sets of X such that {T (α) : α ∈ Σ} covers X and if αn → α in Σ and xn ∈ T (αn ) for all n ∈ N then {xn }∞ n=1 has a cluster point x ∈ T (α). Let H be a functionally bounded relatively sequentially complete set in C p (X ), whose closure υC p (X ) H in υC p (X ) we shall represent by K . As H is functionally bounded in C p (X ), clearly K is a compact subset of υC p (X ), [84, 4.7 Proposition]. Note that each δx ∈ L (X ) with x ∈ X is a σ (C (X ) , L (X ))-continuous linear form on C (X ). Denote by δxυ the (unique) continuous extension of δx to the Hewitt realcompactification υC p (X ) of C p (X ) and define

Sα = { δxυ K : x ∈ T (α)} ⊆ C (K ) for each α ∈ Σ. We claim that Sα is a compact subset of C p (K ).

Let us show in first place that Sα is countably compact. If { δxυn K : n ∈ N} is a sequence in Sα there are x ∈ T (α) and a subnet {yd : d ∈ D} of {xn }∞ n=1 such T under the relative topology of X , so that f that yd → x in → f (x) or (α) (y ) d  rather δ yd , f → δx , f  for all f ∈ C (X ). Hence, for each u ∈ υC p (X ) there is f u ∈ C (X ) with δxυn (u) = δxn ( f u ) for every n ∈ N (see [94, Lemma 9.1]). So, using   that δ yd , f → δx , f  for every f ∈ C (X ), it follows that δ υyd (u) → δxυ (u) for all



u ∈ υC p (X ). In particular, δ υyd K (u) → δxυ K (u) for every u ∈ K , which means



that δ υyd K → δxυ K on Sα under the relative topology of C p (K ). This shows that Sα is a countably compact subspace of C p (K ). But, given that K is compact, C p (K ) is angelic by virtue of the classic Grothendieck theorem [133, Section 1, Theorem 3]. So we conclude that Sα is compact.

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Set M := ∪ {Sα : α ∈ Σ} ⊆ C (K ). We  claim that M is a Lindelöf Σ-subspace of C p (K ). Define the mapping S : Σ → K C p (K ) by the

rule S (α) = Sα . If αn → α in Σ and h n ∈ S (αn ) for each n ∈ N, then h n = δzυn K for some z n ∈ T (αn ) and ∞ n ∈ N. Let z ∈ T (α) be a cluster of thesequence

that δz is a  point  {z n }n=1 in X , so ∞ cluster point of the sequence δzn n=1 in C p C p (X ) . Setting h := δzυ K ∈ S (α), it can be shown as before that h is a cluster point of {h n }∞ n=1 in C p (K ) belonging to S (α). So, M is a Lindelöf Σ-subspace of C p (K ), as stated. Next we claim that M separates the points of K . Otherwise there are u = v in K such that h (u) = h (v) for every h ∈ M. This means that δxυ (u) = δxυ (v) for every x ∈ X . If { f d : d ∈ D} and {gd : d ∈ D} are nets in H with f d → u and gd → v in K then ( f d − gd ) (x) → δxυ (u) − δxυ (v) = 0, so that f d − gd → 0 in C p (X ). Hence f d − gd → 0 in υC p (X ), which yields u − v = 0. Since K is compact and M a Lindelöf Σ-subspace of C p (K ) that separates the points of K , [129, Theorem 3.4] (see also [41, Theorem 91]) ensures that C p (K ) is a Lindelöf Σ-space. Hence K is a Gul’ko compact subset of υC p (X ), so a FréchetUrysohn space (see for instance Lemma  2]).  [39, υC p (X ) ⊆ C (X ). Indeed, if u ∈ K then u ∈ Finally we claim that K := H υC p (X )

H , closure in υC p (X ). Thus, there is a sequence { f n }∞ n=1 in H such that is a Cauchy sequence in C f n → u in υC p (X ). Since { f n }∞ p (X ) and H is relan=1 tively sequentially complete in C p (X ), it follows that u ∈ C (X ). Thus H is relatively compact in C p (X ). Although every locally convex μ-space C p (X ) enjoys the property stated in the previous theorem, the converse statement does not hold, as the following example shows. Example 9 If X is a Lindelöf Σ-space space then C p (X ), though angelic, need not be a μ-space. Let Z be the Reznichenko compact space mentioned in [3, Example 7.14]. This is a Talagrand compact space with a nonisolated point p such that Z = βY with Y = Z \ { p}. Hence Y is a pseudocompact not realcompact space, so that Z = υY . So, C p (Y ) is a continuous image of C p (Z ). This shows that C p (Y ) is K -analytic, which implies that C p (X ) is angelic if X := C p (Y ). Observe that Y is (homeomorphic to) a closed functionally bounded set in C p (X ) which is not compact. Consequently C p (X ), though angelic, is not a μ-space. Of course, Y is not sequentially complete in C p (X ). Otherwise it would be compact by virtue of Theorem 34. Corollary 3 (Baturov [8]) If X is a Lindelöf Σ-space, every countably compact set in C p (X ) is a monolithic compact set. Corollary 4 (Orihuela [120, Theorem 3]) If X is a Lindelöf Σ-space, C p (X ) is angelic. According to [41] a family {Uα,n : (α, n) ∈ NN × N} of closed subsets of X is called a framing if (i) for each α ∈ NN the layer {Uα,n : n ∈ N} is an increasing covering of X , and (ii) for every n ∈ N one has that Uβ,n ⊆ Uα,n , α ≤ β.

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´ Theorem 35 (Ferrando-Kakol-Sliwa [67, Theorem 13]) The space C p (X ) has a bounded resolution if and only if there exists a framing {Uα,n : (α, n) ∈ NN × N} in X enjoying the property that if f ∈ C(X ) there exists γ ∈ NN such that | f (x)| ≤ n for each x ∈ Uγ,n and n ∈ N. A Tychonoff space X admits a fundamental resolution of functions if there exists on X a family of nonnegative real-valued functions { f α : α ∈ NN } such that f α ≤ f β for α ≤ β and for each f ∈ C(X ) there exists α f ∈ NN with | f | ≤ f α f . If X admits a framing as stated in Theorem 35, this is called a nice framing of X . ´ Corollary 5 (Ferrando-K¸akol-Sliwa [67, Corollary 14]) A Tychonoff space X has a fundamental resolution of functions if and only if C p (X ) has a bounded resolution, if and only if X has a nice framing. A Tychonoff space X is called strongly projectively σ-compact if every continuous metrizable image of X is σ-compact. It turns out that X is strongly projectively σcompact if and only if X is projectively σ-compact [67, Corollary 7]. ´ Theorem 36 (Ferrando-K¸akol-Sliwa [67, Theorem 15]) A metrizable space X is σ-compact if and only if X admits a nice framing. The same holds if X is cosmic. Proof The proof follows directly from Theorems 23 and 35. ´ Theorem 37 (Ferrando-K¸akol-Sliwa [67, Theorem 16]) If X has a nice framing, C p (X ) is K -analytic-framed in R X and angelic. Proof The first statement follows from Theorems 21 and 35. Then Theorem 20 shows that υ X is a Lindelöf Σ-space and Orihuela’s angelicity applies to get that C p (X ) is angelic. One may expect that each nice framing for a separable and metrizable X should contain a layer consisting of compact sets, so providing a σ-compact cover of X . We prove however the following property holds. ´ Theorem 38 (Ferrando-K¸akol-Sliwa [67, Theorem 24]) If M is the class of metrizable and separable spaces with a nice framing, then 1. Each X ∈ M, admits a nice framing such that for each α ∈ NN the layer {Uα,n : n ∈ N} consists of compact sets. 2. If X ∈ M is non-Polish, then X admits also a nice framing such that for each α ∈ NN there exists n ∈ N such that Uα,n is not compact. If C p (X ) is K -analytic framed in R X , equivalently, if C p (X ) admits a bounded resolution (Theorem 21), we know by the Arkhangel’ski˘ı-Calbrix theorem that X must be projectively σ-compact. Conversely, we saw earlier in Example 8 that there exists a projectively σ-compact Lindelöf space X such that C p (X ) is not K -analytic framed in R X . Now we have the following example, originally due to Leiderman [106].

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´ Example 10 (Ferrando-K¸akol-Sliwa [67, Example 35]) There exists a Lindelöf space X with a nice framing such that C p (X ) is even K -analytic (so X is projectively σ-compact) but X is not σ-compact. If X is a compact space and Y is a set in the dual C (X )∗ that separates the functions of C (X ), let us denote by σY the weak topology σ (C (X ) , span (Y )) on C (X ), where span (Y ) stands for the linear span of Y . If Y = X , clearly σY = τ p |C(X ) . Let us recall a classic result of Talagrand. Theorem 39 (Talagrand [129, Theorem 3.4]) If X is compact, then C p (X ) is K analytic if and only if C (X ) is weakly K -analytic. Thus X is Talagrand compact space if and only if C (X ) is weakly K -analytic. The next theorem sharps the previous one. Theorem 40 (Ferrando-K¸akol-López-Pellicer [59, Theorem 4.1]) Let X be a compact space and Y be a G δ -dense subspace of X . Then X is a Talagrand compact set if and only if the space (C (X ) , σY ) is K -analytic.

6 Research on C k -theory First note that an analogous to Theorem 19 holds for the space Ck (X ). Theorem 41 (Ferrando-Moll [78, Theorem 3]) The following are equivalent 1. Ck (X ) is a quasi-Souslin space. 2. Ck (X ) is K -analytic. Proof If Ck (X ) has a resolution consisting of compact sets, so does C p (X ). So, Lemma 2 and Theorem 21 ensure that υ X is a Lindelöf Σ-space and C p (X ) is angelic. Therefore Ck (X ) is angelic as well [84, 3.3 Theorem]. Since Ck (X ) is a quasi-Suslin space, necessarily Ck (X ) must be K -analytic [15]. A locally convex space E is said to have a G-base [94, Chap. 1] if there exists a basis {Uα : α ∈ NN } of (absolutely convex) neighborhoods of the origin in E such that Uβ ⊆ Uα whenever α ≤ β. Theorem 42 (Ferrando-K¸akol [48, Theorem 2]) Ck (X ) has a G-base if and only if X has a compact resolution that swallows all compact sets. Proof For each compact K ⊆ X and > 0 define   [K , ] = f ∈ C (X ) : supx∈K | f (x)| ≤ . If {Aα : α ∈ NN } is a compact resolution of X , set Uα = [Aα , α (1)−1 ] for α ∈ NN and put U = {Uα : α ∈ NN }. Clearly U is a family of absolutely convex and absorbing

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sets in C(X ) such that Uβ ⊆ Uα whenever α ≤ β, composing a filter base. The reader may easily check that U is a G-base for a locally convex topology τ on C (X ) with τ p ≤ τ ≤ τc . Now assume that {Aα : α ∈ NN } swallows all compact sets and let V be a neighborhood of the origin of Ck (X ). If K is a compact set in X with [K , ] ⊆ V for some > 0, choosing γ ∈ NN such that K ⊆ Aγ and γ (1)−1 < then Uγ ⊆ [K , ] ⊆ V . This shows that τ = τc , so {Uα : α ∈ NN } is a G-base for Ck (X ). Conversely, suppose that Ck (X ) has a G-base {Uα : α ∈ NN }. For every set U in C (X ) define a corresponding set U ♦ in X by writing U ♦ = {x ∈ X : | f (x)| ≤ 1 ∀ f ∈ U }. Clearly U ♦ is closed in X and U ⊆ V implies that U ♦ ⊇ V ♦ . If K is compact and > 0 then [K , ]♦ ⊆ K , since if x ∈ X \K there is f ∈ C (X ) with f (x) = 2 and f (K ) = {0}, so that f ∈ [K , ] and x ∈ / [K , ]♦ . If K is compact and 0 < ≤ 1 ♦ ♦ then K ⊆ [K , ] , hence [K , ] = K . In addition, if U is a neighborhood of the origin in Ck (X ) then U ♦ is compact. Indeed, if K is a compact set in X such that [K , ] ⊆ U for some > 0, by the previous observations U ♦ ⊆ [K , ]♦ ⊆ K and hence U ♦ is a closed set of a compact set. Now for any K compact in X there is some Uα ⊆ [K , 1], which means that K = [K , 1]♦ ⊆ Uα♦ . This shows that the family A = {Uα♦ : α ∈ NN } swallows all compact sets, so particularly it covers X . As in addition Uα♦ ⊆ Uβ♦ for α ≤ β, it follows that A is a compact resolution of X . Example 11 Every Polish space has a compact resolution that swallows the compact sets, and every metrizable locally convex space has a G-base. Since by Arens’ classic theorem Ck (X ) is metrizable if and only if X is hemicompact, the fact that NN is not σ-compact ensures that Ck (NN ) is a non-metrizable locally convex space with a G-base. Corollary 6 If Ck (X ) has a G-base, then υ X is K -analytic. Proof If Ck (X ) has a G-base, Theorem 42 provides a compact resolution for X . Hence X is a quasi-Suslin space and [35, Lemma 29] ensures that υ X is K -analytic. A space X is said to be strictly angelic [90] if it is angelic and all separable compact subsets of X are first countable. We denote by τ p , τw , τk and τb the pointwise topology of C (X ), the weak topology of Ck (X ), the compact-open topology of C (X ), and the bounded-open topology of C (X ), respectively. The following theorem classifies some locally convex metrizable topologies on C (X ) in terms of the topology of X , in the line of the preceding theorem. Theorem 43 (Ferrando-Gabriyelyan-K¸akol [44, Theorem 3.1]) If X is a Tychonoff space, the following properties hold 1. There exists a metrizable locally convex topology T on C (X ) such that τ p ≤ T ≤ τk if and only if X is a σ-compact space.

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2. There exists a metrizable locally convex topology T on C (X ) such that τk ≤ T ≤ τb if and only if there is an increasing sequence {An : n ∈ N} of functionally bounded subsets of X swallowing the compact sets of X . 3. There exists a metrizable locally convex topology T on C (X ) such that τ p ≤ T ≤ τb if and only if there is an increasing sequence {An : n ∈ N} of functionally bounded subsets of X covering X . Equivalently, if and only if υ X is σ-compact. 4. There is a metrizable locally convex topology T on C(X ) such that τ p ≤ T ≤ τw if and only if X is countable. 5. There is a locally convex topology T on C (X ) with a G-base such that τ p ≤ T ≤ τk if and only if X has a compact resolution. 6. There is a locally convex topology T on C (X ) with a G-base such that τk ≤ T ≤ τb if and only if X has a functionally bounded resolution swallowing the compact sets. 7. There exists a locally convex topology T on C(X ) with a G-base such that τ p ≤ T ≤ τb if and only if X has a functionally bounded resolution. Equivalently, if and only if υ X is K -analytic. In this case (C (X ) , τb ) is strictly angelic. 8. There is a locally convex topology T on C(X ) with a G-base such that τ p ≤ T ≤ τw if and only if X is countable. Proof To prove (i) and (v), for every set U in C (X ) define U ♦ in X as we did earlier by U ♦ = {x ∈ X : | f (x)| ≤ 1 ∀ f ∈ U }. Clearly, U ♦ is closed in X and U ⊆ V implies that U ♦ ⊇ V ♦ . One may reason as in the proof of Theorem 42 to show the following Claim Let T be a locally convex topology on C(X ) such that τ p ≤ T ≤ τk . If U is a neighborhood of the origin in (C(X ), T ), then U ♦ is compact.  (i) If X = {K n : n ∈ N} is σ-compact with K n ⊆ K n+1 for each n ∈ N, we set

Vn :=

1 f ∈ C (X ) : sup | f (x)| < n x∈K n



Then {Vn : n ∈ N} is an open decreasing base of absolutely convex neighborhoods of the origin of a metrizable locally convex topology T on C (X ) such that τ p ≤ T ≤ τk . Conversely, assume that C (X ) has a metrizable locally convex topology τ p ≤ T ≤ τk with a decreasing base {Un : n ∈ N} of neighborhoods of the origin. Then, by the claim, the family K = {Un♦ : n ∈ N} consists of compact subsets of X . Moreover, if y ∈ X since τ p ≤ T , there exists m ∈ N such that Um ⊆ { f ∈ C (X ) : | f (y)| ≤ 1}, which means that Um ⊆ [{y} , 1], so that y ∈ Um♦ . Thus K is a covering of X and we are done. (v) If X has a compact resolution {K α : α ∈ NN } set  Vα :=

f ∈ C (X ) : sup | f (x)| < x∈K α

1 . α (1)

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Then the family {Vα : α ∈ NN } is an open G-base of a locally convex topology T on C(X ) consisting of absolutely convex sets such that τ p ≤ T ≤ τk . Conversely, assume that C(X ) has a locally convex topology τ p ≤ T ≤ τk with a Gbase {Uα : α ∈ NN }. Then, by the claim, the family K = {Uα♦ : α ∈ NN } consists of compact subsets of X . If y ∈ X since τ p ≤ T , there exists β ∈ NN such that Uβ ⊆ { f ∈ C (X ) : | f (y)| ≤ 1}. With the notation of the proof of Theorem 42, this means that Uβ ⊆ [{y} , 1], so that {y} = [{y} , 1]♦ ⊆ Uβ♦ . Thus y ∈ Uβ♦ , which shows that K is a compact resolution of X . (ii) and (vi) follow from [44, Corollary 2.3]. (iii) It is known [3, III.2.21 Proposition] that X has a sequence of functionally bounded sets covering X if and only if υ X is σ-compact. Now the assertion follows from the first statement of [44, Corollary 2.3]. (iv) follows from (viii). (vii) Note that υ X is K -analytic if and only if υ X has a compact resolution [35, Lemma 29], then observe that {K α : α ∈ NN } is a compact resolution in υ X if and only if {X ∩ K α : α ∈ NN } is a functionally bounded resolution in X . So, if X has a functionally bounded resolution, the space C(X ) admits a locally convex topology T with a G-base such that τ p ≤ T ≤ τb by the second statement of [45, Corollary 2.3] applied to the family S of finite subsets of X . Assume conversely that C(X ) admits a locally convex topology T with a G-base {Vα : α ∈ NN } such that τ p ≤ T ≤ τb . Denote by E the topological dual space of (C(X ), T ). Then the family {Vα0 : α ∈ NN }, the polars being taken with respect to E, covers E and is a resolution of E consisting of absolutely convex σ(E, C(X ))-compact sets. Then the space L p (X ) := (L(X ), σ(L(X ), C(X ))) is a subspace of (E, σ(E, C(X ))). Since every functionally bounded subset of a locally convex space is bounded, we get that {L(X ) ∩ Vα0 : α ∈ NN } is a bounded resolution in L p (X ). Thus υ X is K -analytic by [35, Lemma 30]. (viii) The ‘if’ case is trivial. For the ‘only if’ case, suppose that such T exists and proceed by contradiction by assuming that X is uncountable. Let U = {Uα : α ∈ NN } be a G-base of neighborhoods of the origin of T . Clearly, each Uα ∈ U is a neighborhood of the origin for the weak topology τw . So the family M = {Uα0 : α ∈ NN } (polars in Ck (X ) ) is an NN -increasing family of subsets of F := Ck (X ) consisting of τw -equicontinuous sets. But since T is stronger than τ p , M covers the linear subspace L(X ) of F. Note that each Uα0 is contained in a finite-dimensional subspace of F. Consequently, we have an NN -increasing family M of subsets of F, covering L(X ), consisting of finite-dimensional sets. This implies that each of those sets Uα0 meets the canonical copy δ (X ) of X in (F, σ(F, C(X )) in a finite set Uα0 ∩ δ (X ) (otherwise Uα0 , would be infinite-dimensional due to the fact that δ (X ) is a linearly independent set in F). Hence M meets δ (X ) in a resolution consisting of finite sets. But since X is uncountable, some of these sets must be infinite by [94, Proposition 3.7]. This contradiction shows that X is countable. Let us point out that for a metrizable σ-compact space X , the following results hold.

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Theorem 44 (Gabriyelyan-K¸akol [86, Corollary 2.10]) Let X be metrizable. Ck (X ) has a resolution of compact sets that swallows the compact sets if and only if X is σ-compact. Corollary 7 (Ferrando [40, Proposition 3]) Let X be a metrizable space. Ck (X ) has a fundamental bounded resolution if and only if X is σ-compact. Proof If X is σ-compact, Theorem 44 ensures that Ck (X ) has a resolution consisting of compact sets that swallows the compact sets. So, Ck (X ) has a bounded resolution {Aα : α ∈ NN } consisting of closed absolutely convex bounded sets. As X is a kR -space, Ck (X ) is complete and consequently each Aα is a Banach disk. So, Theorem 29 provides a resolution {Aα : α ∈ NN } for Ck (X ) consisting of Banach disks that swallows the Banach disks, hence the bounded sets in Ck (X ). Thus, Ck (X ) has a fundamental bounded resolution. The converse comes from Corollary 2. Compare the previous results with the following classic theorem. Theorem 45 (Christensen [24, Theorem 3.7]) Let X be a separable metric space. Ck (X ) is analytic if and only if X is σ-compact. A family F of functions from a uniform space (X, N ) into a uniform space (Y, M) is called uniformly equicontinuous [10, X.2.1 Definition 2] if for each V ∈ M there is U ∈ N such that ( f (x) , f (y)) ∈ V whenever f ∈ F and (x, y) ∈ U . A uniform space (X, N ) is called trans-separableif for each vicinity N of N there is a countable subset Q of X such that N [Q] = x∈Q U N (x) = X , where U N (x) = {y ∈ X : (x, y) ∈ N }, see [94, Sect. 6.4]. Equivalently, (X, N ) is trans-separable if each uniform cover of X admits a countable subcover [92]. The word trans-separable was coined by Lech Drewnowski in [30]. We shall require the following two results, the first of them being a characterization of trans-separable spaces. Theorem 46 (Ferrando-K¸akol-López Pellicer [56, Theorem 1]) A uniform space (X, N ) is trans-separable if and only if every pointwise bounded uniformly equicontinuous set of functions from (X, N ) into R is metrizable in Ck (X, τN ). The following result characterizes those Ck (X ) spaces whose compact sets are metrizable in terms of a particular uniformity on X . Theorem 47 (Ferrando [37, Theorem 3]) The compact sets of Ck (X ) are metrizable if and only if (X, M), where M is the uniformity on X generated by the pseudometrics d A (x, y) = sup f ∈A | f (x) − f (y)|

(1)

for each compact set A of Ck (X ), is trans-separable. Proof Let E be the topological dual of Ck (X ). Let us denote by K (Ck (X )) the family of all compact sets of Ck (X ) and by ρ (E, Ck (X )) the locally convex topology

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on E of uniform convergence on all compact sets of Ck (X ). If δ stands for the canonical homeomorphic embedding of X into L p (X ), note that δ(X ) ⊆ L (X ) ⊆ E and observe that the topology ρ X = δ −1 (ρ (E, Ck (X ))) on X is stronger than the original topology of X , since ρ (E, Ck (X ))|δ(X ) is stronger than the topology induced on δ (X ) by the weak* topology σ (E, C (X )) of E. This latter fact implies that C (X ) ⊆ C (X, ρ X ) algebraically. Assuming that all compact sets of Ck (X ) are metrizable, then the topological dual E of Ck (X ) equipped with the locally convex topology ρ (E, Ck (X )) is transseparable, [55, Theorem 2]. This topology ρ (E, Ck (X )) generates a unique admissible translation-invariant uniformity N on E such that τN = ρ (E, Ck (X )). Setting  f instead of f ∈ C (X ) when f is considered as a linear functional on E, observe that (u, v) ∈ N ∈ N with u, v ∈ E if and only if there are A ∈ K (Ck (X )) and > 0 f , u − v < . Particularly, the relative uniformity Mδ of N on such that sup f ∈A   δ (X ) satisfies that   f , δx  −   f , δ y  |< . δx , δ y ∈ N ∩ (δ (X ) × δ (X )) ⇔ sup f ∈A |   This defines a uniformity M on X such that (x, y) ∈ M ∈ M if and only if there are A ∈ K (Ck (X )) and > 0 with sup f ∈A | f (x) − f (y)| < . Given that (X, M) and (δ (X ) , Mδ ) are clearly uniformly isomorphic and, as mentioned before, the class of trans-separable spaces is hereditary and closed under uniform continuous images, it follows that the uniform space (X, M) is trans-separable. Assume conversely that (X, M) is trans-separable when M is the uniformity on X generated by the pseudometrics d A (x, y) = sup f ∈A | f (x) − f (y)| for every A ∈ K (Ck (X )) and let A be a fixed compact subset of Ck (X ). As observed above, A is contained in C (X, τM ) = C (X, ρ X ). Moreover, A is a (pointwise bounded) uniformly equicontinuous set of functions from (X, M) to R since, given > 0, if d A (x, y) < obviously | f (x) − f (y)| < whenever f ∈ A. Consequently, by Theorem 46, the set A is metrizable in Ck (X, τM ). Since τM = ρ X is stronger than the original topology on X , there are as many compact sets in X as in (X, τM ) or more, which ensures that the topology on A inherited from Ck (X ) is stronger than the corresponding one inherited from Ck (X, τM ). Since A is compact in Ck (X ), both topologies coincide and A is metrizable. A topological space satisfies the Discrete Countable Chain Condition (DCCC for short) [143] if every discrete family of open sets is countable. Separable spaces, countably compact spaces and Lindelöf spaces satisfy the DCCC. Further, each paracompact space that satisfies the DCCC is Lindelöf. Hence every metrizable space with the DCCC is separable. Theorem 48 (Cascales-Orihuela [20, Theorem 4]) A topological space satisfies the DCCC if and only if each pointwise bounded equicontinuous set in C (X ) is τ p metrizable. As follows from Theorems 46 and 48, every uniform space (X, N ) such that (X, τN ) has the DCCC is trans-separable.

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Theorem 49 (Ferrando [37, Theorem 4]) If X is a completely regular k-space, the following are equivalent 1. X satisfies the DCCC 2. Every admissible uniformity on X is trans-separable 3. All compact sets of Ck (X ) are metrizable. Proof 1 ⇒ 2. Let N be an admissible uniformity on X , so that the uniform topology τN coincides with the original topology of X . Let D be the family of all uniformly continuous pseudometrics on X . If d ∈ D and Vd, := {(x, y) ∈ X × X : d (x, y) < }, then Vd, ∈ N so that the identity map id : (X, τN ) → (X, d) is continuous. Hence if (X, τN ) satisfies the DCCC, the pseudometric space (X, d) is separable. If Q d is a countable dense subset of (X, d), clearly Vd, [Q d ] = X . Since for each N ∈ N there is d ∈ D such that Vd,1/4 ⊆ N (see [100, Chap. 6]), it follows that N [Q d ] = X . Hence (X, N ) is trans-separable. 2 ⇒ 3. Let us assume that every admissible uniformity on X is trans-separable. Let M be the uniformity on X generated by the pseudometrics (1) when A runs over the compact sets of Ck (X ). We claim that X and (X, τM ) have the same compact sets. First note that, if δ and E denote respectively the canonical embedding of X into L p (X ) and the topological dual of Ck (X ), then (X, τM ) is homeomorphic to δ (X ) when equipped with the relative topology of ρ (E, Ck (X )) on E of uniform convergence on all compact sets of Ck (X ). If K is a compact set of X , then the set K ♦ := { f ∈ C (X ) : supx∈K | f (x)| ≤ 1} is a neighborhood of the origin in Ck (X ) and, consequently, the absolutely convex set



f , u ≤ 1} K ♦0 := {u ∈ E : sup f ∈K ♦   is a weak* compact subset of E, that is, a σ (E, C (X ))-compact set. Since ρ (E, Ck (X )) coincides with σ (E, C (X )) on the Ck (X )-equicontinuous subsets of E, it follows that K ♦0 is a ρ (E, Ck (X ))-compact subset of E. Now, since clearly δ (K ) is a weak* closed subspace of K ♦0 , we have that δ (K ) is ρ (E, Ck (X ))compact. Thus K is a τM -compact space. On the other hand, if K is a τM -compact set of X , since τM is stronger than the original topology of X , the set K is compact in X . If X is a k-space, the claim we have just proved ensures that the uniform topology τM coincides with the original topology of X , so that M is an admissible uniformity on X . Since according to our assumptions (X, M) is trans-separable, an application of Theorem 47 guarantees that every compact set of Ck (X ) must be metrizable. 3 ⇒ 1. Assume that every compact set of Ck (X ) is metrizable. Since every pointwise bounded equicontinuous set of C (X ) is contained in a τc -compact set by Ascoli’s theorem, the fact that on the equicontinuous sets of C (X ) the pointwise and the compact-open topology coincide yields that every pointwise bounded equicontinuous set of C (X ) is τ p -metrizable. So Theorem 48 applies to show that X satisfies the DCCC.

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Theorem 50 (Ferrando [38, Theorem 1]) Ck (X ) has a resolution consisting of equicontinuous sets if and only if there exists an admissible uniformity for X , larger than or equal to the Nachbin uniformity, with a G-base. Proof Assume N is a uniformity for X which contains the Nachbin uniform structure N and let {Uα : α ∈ NN } be a G-base of N . If {αn }∞ n=1 is a sequence in N , encode ∞ {αn }n=1 in α as indicated in [41, p. 107] and define Pα = { f ∈ C (X ) : sup(x,y)∈Uαn | f (x) − f (y)| ≤

1 n

∀n ∈ N}.

We claim that {Pα : α ∈ NN } is a resolution for Ck (X ) consisting of equicontinuous sets. In fact, since if α ≤ β then αn ≤ βn for every n ∈ N, clearly Pα ⊆ Pβ . On the other hand, if f ∈ C (X ), since N is larger than the Nachbin uniformity, f is N -uniformly continuous on X . Bearing in mind that {Uα : α ∈ NN } is a G-base of N , for each n ∈ N there exists αn ∈ NN such that | f (x) − f (y)| ≤ 1/n whenever (x, y) ∈ Uαn , which shows that f ∈ Pα for α defined as above. Finally, let us see that each set Pα is equicontinuous. Indeed, given > 0 take n ∈ N such that 1/n < . According to the definition of Pα there is αn ∈ NN , which we extract from α as explained earlier, such that | f (x) − f (y)| < whenever (x, y) ∈ Uαn and this happens for every f ∈ Pα , which shows that Pα is uniformly equicontinuous, hence equicontinuous. For the converse, suppose that {Pα : α ∈ NN } is a resolution of Ck (X ) consisting of equicontinuous sets. For each α ∈ NN define Vα = {(x, y) ∈ X × X : sup f ∈Pα | f (x) − f (y)| < α (1)−1 }. If α ≤ β then Pα ⊆ Pβ , which implies that Vβ ⊆ Vα . Let us see that {Vα : α ∈ NN } is a base of some uniformity N for X . First observe that the diagonal Δ (X ) = {(x, x) : x ∈ X } is contained in each Vα , so no Vα is empty. On the other hand, clearly {Vα : α ∈ NN } is a filter-base with Vα−1 = Vα . In addition, if β ∈ NN satisfies that β ≥ α with β (1) ≥ 2α (1) we claim that Vβ ◦ Vβ ⊆ Vα . Indeed, if (x, y) ∈ Vβ ◦ Vβ there is z ∈ X with (x, z) , (z, y) ∈ Vβ . Hence | f (x) − f (z)| < β (1)−1 and | f (z) − f (y)| < β (1)−1 for every f ∈ Pβ . So, | f (x) − f (y)| < 2β (1)−1 ≤ α (1)−1 for all f ∈ Pα ⊆ Pβ , which shows that (x, y) ∈ Vα . Let us check that N is an admissible uniformity for X , i. e., that τN coincides with the original topology of X . Since X is completely regular, it suffices to show that X and (X, τN ) have the same continuous functions. Take f ∈ C (X ), pick an arbitrary point x0 ∈ X and choose > 0. Then select α ∈ NN such that f ∈ Pα and α (1)−1 < . Clearly Vα (x0 ) = {y ∈ X : (x0 , y) ∈ Vα } is a τN -neighborhood of x0 , and since | f (x) − f (y)| < α (1)−1 < for every (x, y) ∈ Vα , we have in particular that | f (x0 ) − f (y)| < for all y ∈ Vα (x0 ). This shows that f is continuous at x0 under τN .

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Assume conversely that f ∈ C (X, τN ) and fix x0 ∈ X and > 0. Then there is α ∈ NN with | f (x0 ) − f (y)| < for every y ∈ Vα (x0 ). But, since Pα is equicontinuous at x0 , there exists a neighborhood V of x0 of the original topology of X such that 1 sup |h (y) − h (x0 )| < α (1) h∈Pα for every y ∈ V . Hence if x ∈ V then suph∈Pα |h (x) − h (x0 )| < α (1)−1 , which according to the definition of Vα means that x ∈ Vα (x0 ). This shows that V ⊆ Vα (x0 ) and thus | f (x0 ) − f (y)| < for all y ∈ V . So f is continuous at x0 under the original topology of X and f ∈ C (X ). Let us finally check that the uniformity N generated by the base {Vα : α ∈ NN } is larger than the Nachbin uniformity. We have to prove that every real-valued continuous function on X is N -uniformly continuous. Now, given f ∈ C (X ) and > 0, taking advantage of the fact that {Pα : α ∈ NN } is a resolution of C (X ), we can choose γ ∈ NN such that γ (1)−1 < and f ∈ Pγ . Consequently, for each (x, y) ∈ Vγ it happens that | f (x) − f (y)| < γ (1)−1 < , which shows that f is N -uniformly continuous, as stated. Corollary 8 (Ferrando [38, Corollary 2]) Let X be a kR -space. If Ck (X ) is K analytic then there exists an admissible uniformity for X , larger than or equal to the Nachbin uniformity, with a G-base. A Fréchet space E is called a Strongly Countably Generated (briefly a  Weakly   SWCG) space if every bounded set in E  , μ E  , E is metrizable. Equivalently, E is a SWCG space if given a base of closed absolutely convex neighborhoods of zero {Un : n ∈ N} with 2Un+1 ⊆ Un for each n ∈ N there exists an absolutely convex weakly compact set K ⊆ E such that for every weakly compact (absolutely convex) set L ⊆ E and every n ∈ N there is α (n) ∈ N with L ⊆ α (n) K + Un [49, Theorem 9]. A Fréchet space  E is called Strongly Weakly K -Analytic (briefly  SWKA)  space  if E, σ E, E  admits a compact resolution that swallows the σ E, E  -compact sets. If E is a Fréchet space with a base of closed absolutely convex neighborhoods of zero {Un : n ∈ N} such that 2Un+1 ⊆ Un for each n ∈ N, a resolution {Aα : α ∈ NN } for E is called weakly compactly generated if there exists an absolutely convex weakly compact set K such that Aα =

∞ 

(α (n) K + Un )

n=1

for every α ∈ NN . Clearly Aα ⊆ Aβ whenever α ≤ β, and the condition imposed to the base implies that each Aα is closed. Hence {Aα : α ∈ NN } is a weakly compact resolution for E, as follows from [49, Claim 6].

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Theorem 51 (Ferrando-K¸akol [49, Theorem 11]) A Fréchet space E is SWCG if and only if E has a weakly compactly generated resolution that swallows the weakly compact sets. Proof Assume that E is a SWCG space, and let {Un : n ∈ N} be a base of closed absolutely convex neighborhoods of the origin such that 2Un+1 ⊆ Un for every n ∈ N. For every α ∈ NN set ∞  Aα := (α (n) K + Un ), n=1

where K is the absolute convex weakly compact set mentioned after the definition of SWCG space. Clearly {Aα : α ∈ NN } is a weakly compactly generated resolution for E. If L ⊆ E is a weakly compact set in E, for each n ∈ N there exists γ (n) ∈ N such that L ⊆ γ (n) K + Un , so that L ⊆ Aγ . Hence {Aα : α ∈ NN } swallows the weakly compact sets of E. Assume conversely that E contains a weakly compactly generated resolution {Aα : α ∈ NN } that swallows the weakly compact sets. Then there exists a weakly compact absolutely convex set Q such that Aα =

∞ 

(α (n) Q + Un )

n=1

for every α ∈ NN . If L is any weakly compact set in E there is γ ∈ NN such that L ⊆ Aγ , hence for each n ∈ N one gets L ⊆ γ (n) Q + Un . So E is a SWCG space. Theorem 52 (Ferrando-K¸akol [49, Theorem 22]) If Ck (X ) is a Fréchet space, the following statements are equivalent 1. Ck (X ) is a SWCG space. 2. Ck (X ) is a SWKA space. 3. X is countable and discrete. Proof Clearly 1 ⇒ 2. Equivalence 2 ⇔ 3 is consequence of Theorem 55. If X is countable and discrete then Ck (X ) = R X is reflexive, so 3 ⇒ 1. Finally, a special type of quasi-G-base has been introduced in [58]. A topological group G is said to have a Σ-base if for some unbounded and directed subset Σ of NN there is a base of neighborhoods {Uα : α ∈ Σ} of e such that Uβ ⊆ Uα whenever α ≤ β with α, β ∈ Σ. Theorem 53 (Ferrando-K¸akol-López-Pellicer [58, Theorem 4]) The following assertions are equivalent 1. There is a compact covering {Aα : α ∈ Σ} of X , with Σ unbounded and directed, such that Aα ⊆ Aβ whenever α ≤ β in Σ, that swallows the compact sets. 2. Ck (X ) has a Σ-base of absolutely convex neighborhoods of the origin.

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Theorem 54 (Ferrando-K¸akol-López-Pellicer [58, Theorem 7]) If X is a separable metrizable space which is not Polish, then Ck (X ) admits a Σ-base of neighborhoods of the origin but it does not admit any G-base. So Ck (Q) has a Σ-base of neighborhoods of the origin, but no G-base.

7 Research on the Weak Topology of C k (X) We denote by Cw (X ) the space Ck (X ) equipped with the weak topology of Ck (X ). Theorem 55 (Ferrando-Gabriyelyan-K¸akol [45, Theorem 1.8]) Cw (X ) has a compact resolution that swallows the weakly compact sets if and only if X is countable and discrete. Proof First we claim that if Ck (X ) has a resolution {Aα : α ∈ NN } consisting of weakly compact sets that swallows the weakly compact sets in Ck (X ), each compact a resolution of compact sets, it is K -analytic by set in X is finite. AsC p (X ) admits  Theorem 18,  so Cp C p (X ) is angelic by Lemma 2. Hence, each compact set of X → C p C p (X ) is Fréchet-Urysohn. If there exists an infinite compact set K in X , then K contains an infinite convergent sequence that, together with its limit, is homeomorphic to a metrizable compact subset Q of β X . Thus, there is a continuous linear extender map ϕ : C p (Q) → C p (β X ), [7]. If S : C p (β X ) → C p (X ) is the restriction map Sg = g| X , the mapping ψ = S ◦ ϕ is a continuous linear extender, i. e., ψ ( f )| Q = f for every f ∈ C (Q). This ensures that the linear map ψ : C(Q) → Ck (X ) (weak), where Ck (X ) (weak) stands for the space Ck (X ) equipped with its weak topology, has closed graph. Since C(Q) is a Banach space and Ck (X ) (weak) has a resolution of compact sets, the closed graph theorem exposed in Theorem 15 ensures that ψ : C(Q) → Ck (X ) (weak) is weakly continuous. A routine procedure shows that the family {ψ −1 (Aα ) : α ∈ NN } is a resolution for the Banach space C(Q) consisting of weakly compact sets. If P is a compact set under the weak topology of Ck (Q), then ψ (P) is a compact set in  Ck (X ) (weak). Hence, there is a γ ∈ NN such that ψ (P) ⊆ Aγ , so that P ⊆ ψ −1 Aγ . This means that {ψ −1 (Aα ) : α ∈ NN } swallows the compact sets of C(Q) (weak). But it is shown in [114] that for compact Q, if the Banach space C (Q) has a resolution of weakly compact sets that swallows the weakly compact sets, Q is finite. So Q must be finite, a contradiction. Finally, since each compact set in X is finite, one has Ck (X ) = Ck (X ) (weak) = C p (X ). So X must be countable and discrete by Theorem 27. For the next theorem we shall require the following well-known property. Fact 56 The canonical bijective copy δ (X ) of X in L (X ) is a discrete space under the norm topology of the Banach space C b (X )∗ , which is coarser than the relative strong topology β (L (X ) , C (X )) on δ (X ).

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Theorem 57 (Ferrando-K¸akol [50, Theorem 2.3]) Bounded sets of Ck (X ) are weakly metrizable if and only if X is countable. Proof Let us denote by F the topological dual of Ck (X ) and let us assume that the bounded sets of Ck (X ) are metrizable under the weak topology of Ck (X ). According to [55, Corollary 5], when equipped with the strong topology β (F, C (X )), the space F becomes a trans-separable locally convex space. In other words, if P denotes the family of gauges of the polars in F of the bounded sets of Ck (X ), then F is transseparable when equipped with the topology defined by the family P. On the other τk stands hand, the linear subspace C b (X ) is dense in Ck (X ), so that, if   for the relative compact-open topology induced on C b (X ) by Ck (X ), the dual of C b (X ) , τk also coincides with F. If Q is the closed unit ball of the Banach space C b (X ), provided with the supremum-norm topology, then Q is a bounded subset of Ck (X ). So, if Q 0 stands for the polar of Q in F and q ∈ P denotes the gauge of Q 0 , by the very definition of trans-separable locally convex space, the pair (F, q) is a separable metric space. In particular, the dual L (X ) of C p (X ), regarded as a linear subspace of F with the relative topology Tq induced by the norm q is separable. Since X is a discrete subspace of the separable metric space (L (X ) , Tq ) by virtue of Fact 56, necessarily the space X must be countable. For the converse, assume that X is countable. In order to show that each bounded set in Ck (X ) is weakly metrizable, it suffices to show that on each bounded set in Ck (X ) the weak topology of Ck (X ) coincides with the pointwise topology of C (X ). So, let B be a bounded set in Ck (X ). Then B is pointwise bounded, hence pointwise metrizable by the previous theorem. Now choose a pointwise compact subset K of B. We claim that K is weakly compact. Since C p (X ) is metrizable the space Ck (X ) is weakly angelic, so we need only to check that K is sequentially compact in the weak topology of Ck (X ). Pick a sequence ∞ ∞ { f n }∞ n=1 in K and a select a subsequence {gn }n=1 of { f n }n=1 that converges pointwise to some g ∈ K . According to the version of the Lebesgue dominated convergence theorem for (discrete) measures of compact support, the sequence {gn }∞ n=1 converges to g in the weak topology of Ck (X ), which shows that K is weakly compact, as claimed. Hence the pointwise topology and the weak topology of Ck (X ) have the same compact sets inside of B. But since B is pointwise metrizable, it is a k–space for the pointwise topology, so that the weak topology and the pointwise topology coincide on B. Therefore B must be weakly metrizable. When X is a Lindelöf Σ-space, the following analogous to Theorem 34 holds. Theorem 58 (Ferrando and López-Alfonso [68, Theorem 2]) If X is a Lindelöf Σspace, each functionally bounded (relatively) sequentially complete set in Cw (X ) is (relatively) weakly compact. Proof Assume that there are a subset Σ of NN and a map T from Σ into K (X ) such that {T (α) : α ∈ Σ} covers X and if αn → α in Σ and xn ∈ T (αn ) for all n ∈ N then {xn }∞ n=1 has a cluster point x ∈ T (α). Let F denote the completion

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of (C (X ) , ρ (C (X ) , L (X ))), where ρ (C (X ) , L (X )) stands for the topology on C (X ) of uniform convergence on the compact sets of L p (X ). Since X is a Lindelöf space, hence realcompact, the space Ck (X ) is barrelled by the Nachbin-Shirota theorem. Hence the compact-open topology τk of C (X ) coincides with the strong topology β (C (X ) , E), where E denotes the topological dual of Ck (X ). This implies that τk = ρ (C (X ) , L (X )) = μ (C(X ), E). Let H be a functionally bounded relatively sequentially complete set in Cw (X ), whose closure under the weak topology σ (F, E) we shall represent by K . Since H is clearly functionally bounded in (F, σ (F, E)) and (F, μ (F, E)) is complete, Valdivia’s theorem [136, Theorem 3] ensures that K is σ (F, E)-compact. Let  δx be the σ (F, L (X ))-continuous linear extension of δx to F and put Sα = { δx | K : x ∈ T (α)} for α ∈ Σ. If { δxd | K : d ∈ D} is a net in Sα there are x ∈ T (α) and a subnet {yh }h∈E such that yh → x in T (α), so that δ yh → δx under σ (L (X ) , C (X )). Since σ (L (X ) , C (X )) and σ (L (X ) , F) coincide on Q α := {δx : x ∈ T (α)} (see δx in Sα under σ (L (X ) , F), which [122, Chap. VI, Corollary 3]), we have  δ yh →    impliesthat δ yh | K , g → δx | K , g for every g ∈ K . So, Sα is compact in C p (K ). Set M := {Sα : α ∈ Σ} and define S : Σ → K C p (K ) by S (α) = Sα . If αn → α in Σ and yn ∈ T (αn ) foreach n ∈ N, let y ∈ T (α) be a cluster point in X of the ∞ compact in L p (X ) and sequence {yn }∞ n=1 . Since n=1 Q αn is relatively countably  Q compact L p (X ) is a Lindelöf space by [3, 0.5.14 Corollary], ∞ αn is relatively n=1  Q and  δy in L p (X ). Thus σ (L (X ) , C (X )) and σ (L (X ) , F) coincide on ∞ α n n=1 ∞   is a σ (L (X ) , F)-cluster point of {δ yn }n=1 , which implies that δ y | K is a cluster point of { δ x n | K }∞ n=1 in C p (K ). This shows that M is a Lindelöf Σ-subspace of C p (K ). As L (X ) , F is a dual pair, M separates the points of K . So, C p (K ) is a Lindelöf Σspace and K is Fréchet-Urysohn. Consequently, if u ∈ K there is a sequence { f n }∞ n=1 in H such that f n → u under σ (F, E), which implies that { f n }∞ n=1 is a Cauchy sequence in Cw (X ). Since H is relatively sequentially complete in Cw (X ), it follows that f n → g in Cw (X ). Hence u = g ∈ C (X ), which shows that K ⊆ C (X ).

8 Research on the Bidual of C p (X) The bidual M (X ) of C p (X ) equipped with the relative topology of R X has also deserved some attention. Let us denote by R(X ) the linear subspace of the bidual M (X ) of C p (X ) consisting of those functions of finite support [50, p. 1343].  Note that R(X ) , L (X ) is a dual pair. Following Dieudonné and Schwartz [26] a locally convex space E is called distinguished if E is a large subspace of its weak*   words, E is distinguished   if and only if each bounded set in E . In other bidual  E , σ E , E is contained in the σ E  , E  -closure of a bounded set in E. Equivalently, E is distinguished if and only if the strong dual of E (i. e., the topological dual of E endowed with the strong topology) is barrelled, see [102, 8.7.1]. Let us review some relevant results obtained in [50, 52, 66]. If A is a bounded subset of R X , put φ A = sup {| f | : f ∈ A}, where the supremum is with respect to the usual pointwise ordering of R X , i. e., f ≤ g if f (x) ≤ g (x) for every x ∈ X . In other

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words, φ A (x) = sup {| f (x)| : f ∈ A} for all x ∈ X . Ifg ∈ R X , we designate  by Pg the closed and bounded subset of R X defined by Pg = h ∈ R X : |h| ≤ |g| . If A is a bounded set in C p (X ), so is A+ :=

{C (X ) ∩ PφΛ : Λ is a finite subset of A}.

Lemma 4 ([52, 83]) If A is a bounded set in C p (X ), then A+ is a bounded set in C p (X ) whose closure A+ in R X verifies A+ = Pφ A . Proof Since φΛ ∈ C (X ) for finite Λ ⊆ A, routinely C (X ) ∩ PφΛ = PφΛ , so that A+ ⊇ PφΛ . Consequently, on has that A+ =

{PφΛ : Λ is a finite subset of A} = Pφ A

as desired. Let us say that a set M inR X is cofinal if for each g ∈ R X there is f ∈ M with |g| ≤ | f |, and denote by Cof R X the least cardinality of a cofinal set M in R X . We represent by bn (X ) the so-called borne number of X [52, Sect. 4], which is defined as the least cardinality of a fundamental family of bounded sets in C p (X ). Theorem 59 ([51, Theorem 2]) The following conditions are equivalent. 1. The space C p (X ) is distinguished. 2. The set {φ A : A is bounded in C p (X )} is cofinal in R X . 3. C p (X ) is a large subspace of R X , i. e., each bounded set in R X is contained in the closure in R X of a bounded set in C p (X ). 4. The strong dual L β (X ) of C p (X ) carries the strongest locally convex topology. Proof The weak∗ bidual of C p (X ) is (viewed as) a subspace of R X and contains the subspace R(X ) of all finitely supported functions [50, 52, 83]. If g ∈ R X , then E 0 ∩ Pg is a bounded subset of the bidual. If (1) holds, there is a bounded set A in C p (X ) with A ⊇ E 0 ∩ Pg , closure in R X . Thus φ A = φ A ≥ φ E0 ∩Pg = φ Pg = |g|. If (2) holds and B is a bounded set in R X , there is a bounded set A in C p (X ) with φ A ≥ φ B . Clearly, Pφ A ⊇ Pφ B ⊇ B. The set A+ is bounded in C p (X ) with A+ = Pφ A ⊇ B. Now, if (3) holds, the   strong topology  β (L (X ) , C (X )) of L (X )  coincides with β L (X ) , R X = β L (X ) , L (X )∗ , the strongest locally convex topology. Finally, if statement (4) holds then L β (X ) is barrelled, thus C p (X ) is distinguished. Corollary 9 The following statements hold. 1. 2. 3. 4.

  If C p (X ) is distinguished, then bn (X ) ≥ Cof R X . If bn (X ) ≤ |X | then C p (X ) is not distinguished [66, Theorem 2.1]. If Y ⊆ X and C p (X ) is distinguished, C p (Y ) is distinguished [52, Theorem 16]. If |X | = ℵ0 then C p (X ) is distinguished [50, Theorem 3.3].

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5. If X is a continuous injective image of Y and C p (X ) is distinguished, then C p (Y ) is distinguished. Proof Let F be a fundamental family of bounded sets in C p (X ) with |F| = bn (X ). (1) If C p (X ) is distinguished and g ∈ R X , statement (2) of Theorem 59 gives a bounded set A in C p (X ) with φ A ≥ |g|. There exists B ∈ F with  B ⊇ A, so φ B ≥ φ A ≥ |g|. Therefore {φ B : B ∈ F} is cofinal in R X and Cof R X ≤ |F| = bn (X ). (2) If |F| ≤ |X |, there is a set S = {x A : A ∈ F} ⊆ X with x A = x B for distinct A, B ∈ F. Well-define g ∈ R X so that g| X \S = 0 and g (x A ) = φ A (x A ) + 1 for all A ∈ F. If A is bounded in C p (X ), there is B ∈ F with B ⊇ A. We have φ A (x B ) ≤ φ B (x B ) < g (x B ). So φ A  |g| for each bounded A ⊆ C p (X ), contradicting statement (2) of Theorem 59. (3) If C p (X ) is distinguished and A is a bounded set in RY , then Aext := { f ∈ X R : f |Y ∈ A and f | X \Y = 0} is bounded in R X . Statement (3) of Theorem 59 yields a bounded set B in C p (X ) with B R

RX

⊇ Aext . The set B|Y of restrictions is bounded

Y

in C p (Y ) and verifies B|Y ⊇ Aext |Y = A, which proves statement (3) of Theorem 59 holds with Y in place of X . (4) If g ∈ R X and |X | = ℵ0 , then C p (X ) is a dense subspace of the metric space X R , and some sequence S in C p (X ) converges in R X to g. Clearly, S is bounded in C p (X ) and φ S ≥ |g|, so the second statement of Theorem 59 applies. (5) Let ξ be the original topology of X . By assumption there exists on X a stronger topology γ such that (X, γ) is homeomorphic with Y . Since C p (X, ξ) ⊆ C p (X, γ) ⊆ R X , if C p (X, ξ) is large in R X , so is C p (X, γ). For concrete examples of distinguished and nondistinguished C p (X ) spaces, if X is a cosmic space with |X | = 2ℵ0 then C p (X ) is nondistinguished, so C p (R), C p ([0, 1]), C p (Cantor set) are nondistinguished. Also C p (βN), C p (βQ), C p (M) and C p (S), where M and S are the Michael and the Sorgenfrey lines, respectively, are nondistinguished. If X is either countable or discrete then C p (X ) is distinguished, so C p (Q) and C p (D (m)), where D (m) denotes the discrete space of cardinality m, are distinguished. In addition, if X is a scattered Eberlein compact space, then C p (X ) is distinguished. So, if D (m) denotes the one-point compactification of D (m) then C p (D (m) ) is distinguished. Also, C p (X ) is distinguished if X is strongly splitable (see [52, 66]). The following result provides a characterization of distinguished C p (X ) spaces in terms of the topology of X . Theorem 60 (Ferrando-Saxon [83, Theorem 5]) The space C p (X ) is distinguished if and only if for each countable partition {X k : k ∈ N} of X into nonempty pairwise disjoint sets there is a point-finite sequence of open sets {Uk : k ∈ N} with X k ⊆ Uk for each k ∈ N, i. e., such that each point x ∈ X belongs to Un for only finitely many n ∈ N. Proof Assume that C p (X ) is distinguished. According to Theorem 59, C p (X ) is a large subspace of R X , which means that R X = M (X ). Let P be a partition of X into nonempty disjoint sets X n (n ∈ N). Let h be the step function in R X with constant

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value n on each X n . Since h ∈ R X = M (X ) there is a bounded set B in C p (X ) with h ∈ B. Each set   f −1 (n − 1, n + 1) : f ∈ B Q n := is an open set containing X n by definition of h and B. Moreover, when a point x is in Q n , there is some f n ∈ B such that f n (x) > n − 1. Since B is bounded at x, the point x can be in only finitely many Q n . Thus {Q n : n ∈ N} is a point-finite sequence. Assume conversely that the property of the second part of the statement of the theorem holds. Let f ∈ R X \∞ (X ) be given. Then there exists a sequence of positive integers n 1 < n 2 < · · · < n k < · · · such that the sets X 1 , X 2 , . . . , X k , . . . are nonempty and partition X , where X 1 := {x ∈ X : | f (x)| ≤ n 1 } and X k := {x ∈ X : n k−1 < | f (x)| ≤ n k } for k ≥ 2. Let P := {X k : k ∈ N}. By hypothesis there exists a point-finite sequence {Q k : k ∈ N} of open sets in X such that X k ⊆ Q k . If k ∈ N and Δk is a nonempty finite set in X k , choose f k,Δk ∈ C (X ) such that 0 ≤ f k,Δk ≤ n k with f k,Δk identically n k on Δk and 0 off Q k . Let Fk be the collection  pof all such f k,Δk , and let B be the collection of all functions h of the form h = k=1 f k , where p ∈ N and f k ∈ Fk . Since x ∈ X is in only finitely many Q k , we have f k (x) = 0 for all f k ∈ Fk with k sufficiently large, say, for k ≥ m. Thus 0 ≤ h (x) =

p  k=1

f k (x) ≤

m 

f k (x) ≤ n 1 + · · · + n m

k=1

for each h ∈ B, which proves B is bounded in C p (X ). If x ∈ X p for a given p, we may choose f p := f p,{x} ∈ F p and obtain h ∈ B with h (x) =

p 

f k (x) ≥ f p (x) = f p,{x} (x) = n p ≥ | f (x)| .

k=1

Therefore f ∈ Pφ B . Since Pφ B = B + ⊆ M (X ) by virtue of Lemma 4, it follows that f ∈ M (X ). Thus R X = M (X ) and Theorem 59 ensures that C p (X ) is distinguished. In [95] it is shown that C p (X ) is distinguished if and only if X is a Δ-space in the sense of Knight [101], and several applications of this fact are provided. Next let us investigate the behaviour of the weak* bidual M (X ) of C p (X ) under the existence of a bounded resolution. The following well-known fact will be required below.   Lemma 5 The weak topology σ L (X ) , R(X ) of L (X ) induces on δ (X ) the discrete topology. Since R(X ) is a linear subspace of the bidual M (X ) of C p (X ), the set δ (X ) is also discrete under the weak topology σ (L (X ) , M (X )). Theorem 61 (Ferrando [42, Theorem 19]) If X is realcompact, are equivalent 1. The weak* bidual of C p (X ) has a Σ-covering of limited envelope. 2. X is countable.

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Proof 1 ⇒ 2. If the weak* bidual M (X ) of C p (X ) has a Σ-covering of limited envelope, so does the linear subspace R(X ) + C (X ) equipped with the relative topology of the product topology of R X , since R(X ) ⊆ M (X ), [50, page 1343]. Therefore, if we set F := R(X ) + C (X ), Lemma 1 provides a topological subspace Z of R F which is a Lindelöf Σ-space and verifies L (X ) ⊆ Z ⊆ R F . This means exactly that, when we equip the dual L (X ) of F with the weak topology σ (L (X ) , F), it becomes a topological subspace of the Lindelöf Σ-space Z . If τ denotes the relative topology of σ (L (X ) , F) on the copy δ (X ) of X in L (X  ), we claimthat (δ (X ) , τ ) is complete. Let {xd : d ∈ D} be a net in X such that δxd : d ∈ D is a τ -Cauchy net in δ (X ). Let x be any point of X . Since δxd : d ∈ D is a τ -Cauchy net in δ (X ), for 0 < < 1 there exists d ( ) ∈ D such that 0  δxr − δxs ∈ −1 χ{x} .



for r, s ≥ d ( ), that is χ{x} (xr ) − χ{x} (xs ) < for r, s ≥ d ( ). So, either there is a d1 ≥ d ( ) such that xr = x for every r ≥ d1 or such d1 does not exist. Now, let w 0    be the limit of δxd : d ∈ D in the complete space R F . Since 2χ{x} , as a subset 0  F of R F , is closed

in R , it follows  that δxr − w ∈ χ{x} for all r ≥ d ( ), which implies that χ{x} (xr ) − w, χ{x} ≤ for r ≥  d ( ). If there is no d1 ≥ d ( ) such that xr = x for every r ≥ d1 , this means that w, χ{x} ≤ . As we may repeat the same argument with another smaller > 0, we  conclude that if there is no q ∈ D such that x p = x for every p ≥ q, then w, χ{x} = 0. But we may repeat this argument again for all x ∈ X . Hence,  for each x ∈ X , if the  net {xd : d ∈  D} is not eventually equal to x then w, χ{x} = 0. Since the family χ{x} : x ∈ X is a Hamel basis for R(X ) , this implies that if {xd : d ∈ D} is not eventually constant then w|R(X ) = 0. F F  On the other hand, since δxd : d ∈ D converges in R to w ∈ R , then δxd , f → w, f  for every f ∈ F, i.e., f (xd ) → w, f  for every f ∈ F and, in particular, for every f ∈ C (X ). Thus δxd : d ∈ D is a Cauchy net in RC(X ) . Since X is realcompact, it is C (X )-complete (see [89, 15.14 Corollary] or [41, Theorem 24]), which means that δ (X ) is a complete subspace of L p (X ). So, there is x ∈ X such that δxd → δx in L p (X ). This entails that f (xd ) → f (x) for every f ∈ C (X ), which implies that w, f  = δx , f . Therefore w|C(X ) = δx . In other words, if {xd : d ∈ D} is not eventually constant then w acts on F = R(X ) + C (X ) as 0 + δx = δx ∈ δ (X ). So, whether the net {xd : d ∈ D} is eventually constant or not, we conclude that it converges to a point of δ (X ). Consequently, δ (X ) is a complete set in (L (X ) , σ (L (X ) , F)), as stated. All this means that (δ (X ) , τ ) is a closed subspace of the complete space R F . Hence, a closed subspace of the Lindelöf Σ-space Z . Therefore, (δ (X ) , τ ) is also a Lindelöf Σ-space, hence a Lindelöf space. Since (δ (X ) , τ ) is discrete by Lemma 5 and each discrete Lindelöf space is countable, we are done. 2 ⇒ 1. If X is countable then C p (X ) is metrizable. So, if {Un : n ∈ N} is adecreasing base of absolutely convex neighborhoods of the origin, the family 00 N { ∞ n=1 α (n) Un : α ∈ N } is a resolution of absolutely convex weak* compact sets for the bidual M (X ) of C p (X ) that swallows the Banach disks of M (X ). Hence,

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if C p (X ) is distinguished that family swallows the weak* compact sets of M (X ), so the bidual of C p (X ) has a resolution of absolutely convex weak* compact sets, hence a Σ-covering of limited envelope. Remark 2 It is useful to point out that the argument of the proof of Theorem 61 actually shows that if X is realcompact and the linear subspace R(X ) + C (X ) of the weak* bidual of C p (X ) has a Σ-covering of limited envelope, then X must be countable. Corollary 10 (Ferrando [42, Corollary 22]) Let X be realcompact. The weak* bidual M (X ) of C p (X ) is a Lindelöf Σ-space if and only if X is countable. Proof If the weak* bidual M (X ) of C p (X ) is a Lindelöf Σ-space, by Theorem 20 it has a Σ-covering of limited envelope, so Theorem 61 applies. Conversely, if X is countable then E of coincides with R X by [50, Theorem 3.3]. So, in this case M (X ) is linearly homeomorphic to RN , which is analytic, hence K -analytic. Theorem 62 (Ferrando [42, Theorem 28]) The weak* bidual of C p (X ) has a resolution consisting of pointwise bounded sets if and only if X is countable. Proof If the weak* bidual M (X ) of C p (X ) has a resolution consisting of weak* bounded sets, [42, Lemma 23] asserts that the linear subspace R(υ X ) + C (υ X ) of M (υ X ) has a resolution consisting of Rυ X -bounded sets, hence a Σ-covering of limited envelope. Since υ X is realcompact, Remark 2 guarantees that υ X is countable. So, X is countable. The converse statement is a consequence of the first and of C p (X ) metrizability. Example 12 Since R is hemicompact, the space Ck (R) is metrizable. Hence C p (R) has a resolution of bounded sets. However its weak* bidual M (R) of C p (R) lacks such a resolution because of Theorem 62. Notice that Theorem 28 follows as a straightforward consequence of Theorem 62. Acknowledgements My gratitude to Miguel Hernández University and in particular to CIO Institute and to Department of Statistics, Mathematics and Informatics by the organization and support of the International Meeting on Functional Analysis and Continuous Optimization dedicated to Juan Carlos Ferrando on the occasion of his 65 birthday in Elche, Spain, on June 16-17, 2022. I have been very honored by the invitation to participate in this Meeting with a talk on the research work of professor Juan Carlos Ferrando. I also thank the work of professors María Josefa Cánovas Cánovas, José María Amigó García and Marco Antonio López-Cerdá who have organized, developed and managed this International Meeting with my congratulations by this wonderful Meeting. And finally, last but not least, my gratitude to professor Juan Carlos Ferrando by his friendship for almost 40 years, my respect for his enormous work capacity, and my admiration for his amazing sharpness and ability in research. Juan Carlos is a stimulating example for all of us who have known him. I would like to continue working in mathematics with my excellent and admired friend Juan Carlos Ferrando for many more years.

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A Class of Monothetic Reflexive Groups and the Weil Property L. Außenhofer

Dedicated to Prof. Juan Carlos Ferrando

Abstract In this paper we study groups which have the so called Weil property (strong Weil property), which means that all cyclic (finitely generated) subgroups are discrete or precompact (locally precompact). The strong Weil property implies the Weil property but not vice versa. We study permanence properties and observe that a reflexive group need not have the Weil property, and conversely, a group having the strong Weil property need not be reflexive. Next to that, a new class of monothetic reflexive groups which do not have the Weil property is presented. Keywords Weil’s Lemma · Pontryagin reflexive group · Topology of uniform convergence · Completion · Finitely generated subgroup MSC Classification 22A05 · 22B05

1 Introduction All groups in this article are assumed to be abelian and all topological groups are assumed to be Hausdorff groups. For a topological group G the set G ∧ of all continuous homomorphisms into T is called character group of G. With pointwise defined addition and endowed with the compact-open topology G ∧ is a topological group. This allows to form the L. Außenhofer (B) Universität Passau, Innstr. 33, 94932 Passau, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_2

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second character group G ∧∧ . There is a canonical homomorphism αG : G → G ∧∧ which maps x onto the point-evaluation αG (x) : G ∧ → T, χ → χ (x). In case αG is a topological isomorphism, the group G is called Pontryagin reflexive (or only reflexive if no confusion with reflexive vector spaces can arise). This term honours Pontryagin who proved together with van-Kampen that every locally compact abelian (LCA for short) group is Pontryagin reflexive. The next classes of reflexive groups were products of reflexive groups [8], reflexive vector spaces and Banach spaces [11], and free abelian groups over 0-dimensional Dieudonné-complete k-spaces [9]. All these groups have a similar “local structure” in the following sense: Every finitely generated subgroup is locally precompact and every cyclic subgroup is either discrete or precompact. (In the case of LCA groups this is the statement of the famous Weil theorem.) These properties are also stable under taking products (Propositions 3.6 and 3.8). Honouring Weil, we say that a topological group has the Weil property if every cyclic subgroup is either discrete or precompact and that it has the strong Weil property if every finitely generated subgroup is locally precompact. Recall that a group is called monothetic if it has a dense cyclic subgroup. In 1964, Rolewicz proved in [10] that the group c0 (T) of all null-sequences in the torus T endowed with the topology of uniform convergence is monothetic but neither discrete nor compact. Hence this group does not have the Weil property. It was shown by Gabriyelyan in [6, Theorem 1] that c0 (T) is reflexive. In the same article, Gabriyelyan showed that there exists a non-discrete and non-precompact topology on the integers which turns the integers into a reflexive group. This is another example of a group which does not have the Weil property, but which is reflexive. So the Weil property is not necessary for the reflexivity of a group. In this paper, we show that even a complete and metrizable group which has the strong Weil property need not be reflexive (Example 3.5). The strong Weil property implies the Weil property (Proposition 3.3) but not vice versa (Theorem 4.8). Permanence properties of groups having the (strong) Weil property are studied and examples of groups having the (strong) Weil property are given in Sect. 3. Next to that, we present a new class of monothetic reflexive groups which do not have the Weil property: Let b = (bn )n∈N0 be a sequence of natural numbers such that → ∞. Denote by τb the topology b0 = 1 and bn divides bn+1 for all n ∈ N0 and bbn+1 n of uniform convergence on Z on the set { b1n + Z : n ∈ N} ⊆ T. It was shown in [3] that the character group of (Z, τb ) is isomorphic to Z(b∞ ) = { bkn : n ∈ N0 , k ∈ Z}. The character group of the discrete group Z(b∞ ) is known to be isomorphic to the )n∈N0 -adic integers. For the readers’ convenience, we present a group q of q = ( bbn+1 n short explanation of this isomorphism. The fact that (Z, τb )∧∧ can be identified with a subgroup of q enables us to describe the completion C of (Z, τb ) and to show that this group is reflexive. The article is organized as follows: In Sect. 2 we introduce some notation and recall facts about duality theory applied later in this article. In Sect. 3 permanence properties of groups with the (strong) Weil property are studied and some examples are given. In Sect. 4 we start with a short repetition of q-adic integers and prove

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51

afterwards that the completion C of (Z, τb ) is a reflexive group. Finally, in Sect. 5, we define a group topology τ on Z2 such that (Z2 , τ ) has the Weil property but not the strong Weil property and such that its completion does not have the Weil property.

2 Notation and Preliminaries As usual, let N0 = {0, 1, 2, . . .} denote the natural numbers. Let N = N0 \ {0}, and let Z denote the integers, and let T be the compact quotient group R/Z. For m ∈ N, m ≥ 2 the cyclic group of order m is denoted by Z(m) = Z/mZ. The rank of a torsion-free group is defined to be the maximum number of independent elements. Let G be a group and H a point-separating subgroup of the group Hom(G, T) of all (not necessarily continuous) homomorphisms G → T. By σ (G, H ) we denote the topology on G induced by the embedding G → T H , x → (χ (x))χ∈H . 1 1 ≤ x ≤ 4m } and let T+ := T1 . For a For m ∈ N, let Tm := {x + Z ∈ T : − 4m

∧ subset A of a topological group G let A = {χ ∈ G : χ (A) ⊆ T+ } and for B ⊆ G ∧ , let B  := {x ∈ G : ∀χ ∈ B χ (x) ∈ T+ }. A subset A of G is called quasi-convex if A = (A ) holds. A topological group G is said to be locally quasi-convex if there is a neighborhood base at the neutral element 0 consisting of quasi-convex sets. Now we are gathering some facts on duality theory applied in Sect. 4. Facts 2.1 1. ([5], [2, 4.5]) Let H be a dense subgroup of the metrizable group G and let ι : H → G denote the embedding. Then the dual homomorphism ι∧ : G ∧ → H ∧ , χ → χ | H is a topological isomorphism. 2. [2, 6.16] The completion of a locally quasi-convex group is again locally quasiconvex. 3. [2, 5.12,6.10] If G is a locally quasi-convex metrizable group then αG : G → G ∧∧ is an embedding. 4. [4, 13.1.2.(f)] Let G be an abelian topological group. For every neighborhood U of 0 in G, the polar U is a compact subset of G ∧ endowed with the compact-open topology. 5. If G is a metrizable group with neighborhood base (Um )m∈N at 0, then (Um

)m∈N is a neighborhood base at 0 in G ∧∧ . Proof of 5.: Since G is metrizable, αG is continuous [4, 13.4.1]. This means that every compact subset of the character group is equicontinuous [2, 5.10]. The assertion is a consequence of the definition of the compact-open topology. 

3 The (Strong) Weil Property A topological group is called monothetic if it has a dense cyclic subgroup. Necessarily, every monothetic group is abelian. Weil proved that every monothetic LCA group is either compact or discrete. In other words:

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Theorem 3.1 (Weil’s Lemma) [14, Lemma 2, p. 96] Every cyclic subgroup of a locally compact group is either precompact or discrete. We are studying this property in the context of abelian groups: Definition 3.2 An abelian group (G, τ ) is said to have the Weil property if every cyclic subgroup with the induced topology is either discrete or precompact. It is said to have the strong Weil property if every finitely generated subgroup is locally precompact. Observe that every cyclic subgroup of R √ (with the usual topology) is discrete; however, the subgroup generated by 1 and 2 is dense in R and hence locally precompact, but no longer discrete. Proposition 3.3 If a group has the strong Weil property, then it has the Weil property.  the completion Proof Suppose that G has the strong Weil property. Denote by G of G. Let x ∈ G. By assumption, x is locally precompact, hence the closure x  is locally compact. Now Weil’s Lemma applies and shows that x is either in G discrete or compact. In the first case, x = x is discrete, in the latter case x is precompact.  In Example 4.8 we will present a non-locally precompact topology τ on Z2 such that every cyclic subgroup of (Z2 , τ ) is discrete. So this group has the Weil property but not the strong Weil property. Examples 3.4 (a) Obviously, every locally precompact group has the strong Weil property. (b) Every subgroup of a group which has the (strong) Weil property has the (strong) Weil property. (c) Every real or complex topological vector space has the strong Weil property, since every finite-dimensional subspace is isomorphic to Rn or Cn for a suitable n ∈ N0 . (d) Every free abelian group A(X ) has the strong Weil property. [Indeed, this follows from (b) and (c), since A(X ) embeds in the free locally convex space L(X ) by [12, 13].] (e) Every abelian Hausdorff group is a quotient group of a group having the strong Weil property. [This is a consequence of (d), since A(G) → G is a projection for every abelian Hausdorff group G (c.f. [1, 7.1.9].) p

Example 3.5 For 1 < p < ∞, let G = LZ ([0, 1]) be the group of equivalence classes of almost everywhere Z-valued and p-integrable functions on [0, 1]. By [2, 11.15], G is a closed subgroup of L p ([0, 1]) and not reflexive. Since G has the strong Weil property by Examples 3.4 (b) and (c), this shows that even a complete and metrizable group which has the strong Weil property need not be reflexive. We are going to study products of groups having the (strong) Weil property:

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Proposition 3.6 Every product of groups which have the Weil property has the Weil property again.  Proof Let G i be groups having theWeil property. Fix (xi ) ∈ i∈I G i . If all groups xi  are precompact, then (xi ) ≤ i∈I xi  is precompact as well. Suppose now that for some i 0 ∈ I the cyclic group xi0  is not precompact, hence infinite and discrete. Since the projection (xi ) → xi0  is injective and continuous, we conclude that also  (xi ) is discrete. In order to prove that the class of groups having the strong Weil property is closed under taking products, we need the following Lemma. Lemma 3.7 Every subgroup of finite rank of a product of discrete torsion-free groups (endowed with the product topology) is discrete. Proof Let ∅ =  I be an index set and let Di be torsion-free abelian groups for all i ∈ I . Let F ≤ i∈I Di =: G be a subgroup of finite rank. Let πi : G → Di denote the canonical projection. Wlog {0} = πi (F) = Di for all i ∈ I . We are going to prove the assertion by induction on the rank n of the subgroup F of G. If n = 0 then F = {0} and the assertion trivially holds. Assume now that every subgroup of rank ≤ n of G is discrete. Let F be a subgroup of G of rank n + 1. Fix i 0 ∈ I and let F0 = ker(πi0 ). Since {0} is open in Di0 , F0 is an open subgroup of F. So it is sufficient to prove that F0 is discrete. From {0} = πi0 (F) = Di0 , we obtain F0 = F. Since F/F0 ∼ = Di0 is torsion- free, the rank of F0 is strictly smaller than the rank of F and hence the inductive hypothesis applies and shows that F0 is discrete.  Proposition 3.8 Every product of groups which have the strong Weil property has again the strong Weil property.  Proof Let F be a finitely generated subgroup of i=I G i =: G where all groups G i have the strong Weil property. Let πi : G → G i denote the canonical projection. Since for every i ∈ I the subgroup πi (F) of G i is finitely generated, the assumption that G i has the strong Weil property implies that πi (F) is locally precompact. So without loss of generality we may assume that G i = πi (F) is locally precompact. i the completion of G i , which is an LCA group. By the structure theorem Denote by G i ∼ of LCA groups (c.f. [7, (24.30)], [4, 14.2.18]), G = Rni × Hi where n i ∈ N0 and the group Hi has a compact open subgroup K i . Since Hi /K i is a discrete group having a finitely generated dense subgroup, it is itself a finitely generated discrete group. By the structure theorem of finitely generated abelian groups, we may assume wlog that Hi /K i is torsion-free, by replacing K i by a larger compact subgroup if necessary. So  a topological isomorphism, F is a subgroup of  afterniapplying  R × i∈I i∈I Hi =: G. i → Rni and qi : G  Let pi : G  i → Hi denote the canonical projections. First we consider ( pi ◦ πi | F ) : F → i∈I Rni , x → ( pi (πi (x)))i∈I . Since ( pi ◦ πi )(F) is finitely generated, this group is contained in a finite dimensional subspace of the

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 topological vector space i∈I Rni . So after a trivial identification  of ( pi ◦ πi )(F) with a subgroup of Rn for a suitable n ∈ N0 we have F ≤ Rn × i∈I Hi . Next, we consider the composition of the following mappings   ϕ : F −→ −→ i∈I Hi i∈I Hi /K i x −→ (qi (πi (x)))i∈I −→ (qi (πi (x)) + K i )i∈I  Then ϕ induces a continuous injective mapping F/ker(ϕ) → i∈I Hi /K i . All groups Hi /K i are torsion-free and discrete. The finitely generated torsion-free group F/ker(ϕ) has finite rank. So the above Lemma 3.7 implies that ϕ(F) is a discrete group. This implies that also F/ker(ϕ) is discrete or, equivalently, that F0 := ker(ϕ) is an open subgroup of F. So it is sufficient  to prove that F0 is locally precompact.  We have shown that F0 embeds in Rn × i∈I K i . By the Tychonoff theorem, i∈I K i is compact, so F0 is a locally precompact group.  Combining Examples 3.4 (b) and Propositions 3.8 and 3.6 yields:  Corollary 3.9 Let I = ∅ and (G i )i∈I be a family of groups. Then i∈I G i has the (strong) Weil property if and only if all groups G i have the (strong) Weil property.

4 A Class of Monothetic Reflexive Groups 4.1 The q-adic Integers—A Repetition First, we recall the definition of the q-adic integers (cf. [7, (10.2),(10.3)]). Let q = (qn )n∈N0 be a sequence of positive integers greater or equal than 2. For = {0, 1, 2, . . . , qn − 1}. every n ∈ N0 , let Fq n On the set q = n∈N0 Fqn an addition is defined as follows: Let (xn ), (yn ) ∈ q . Choose z 0 ∈ Fq0 and t0 ∈ N0 such that x0 + y0 = z 0 + t0 q0 . Next, choose z 1 ∈ Fq1 and t1 ∈ N0 such that x1 + y1 + t0 = t1 q1 + z 1 . Suppose that z 0 , . . . , z m and t0 , . . . , tm ∈ N0 have already been constructed. Then choose the unique z m+1 ∈ Fqm+1 = {0, . . . , qm+1 − 1} and tm+1 ∈ N0 such that xm+1 + ym+1 + tm = z m+1 + tm+1 qm+1 . In this way (z n )n∈N0 can be constructed inductively. The sum (xn )n∈N0 + (yn )n∈N0 is then defined to be (z n )n∈N0 . Note that the n-th coordinate of the sum of two elements depends only on the first n coordinates of the summands. It is shown in [7, 10.3] that the so-defined addition is well-defined, and that (q , +) is an abelian group.  Put b0 = 1 and for n ∈ N, bn = n−1 k=0 qk . Every natural number k ∈ N0 can be N uniquely written in the form n=0 xn bn where xn ∈ Fqn . Applying this notation, the mapping N  x j b j → (x0 , . . . , x N , 0 . . .)

: N0 → q , k = j=0

A Class of Monothetic Reflexive Groups and the Weil Property

55

is injective and additive. The set 0 := (N0 ) is the submonoid of q which consists of all finite sequences in q . Definition 4.1 A strictly increasing sequence (bn )n∈N0 of natural numbers is called a D-sequence, if b0 = 1 and if bn divides bn+1 for all n ∈ N. The set of all D-sequences is denoted by D. Put D1 := {(bn )n∈N0 ∈ D :

bn+1 n→∞ −→ ∞}. bn

For a D-sequence b = (bn )n∈N0 we define Z(b∞ ) =



 k + Z : k ∈ Z, n ∈ N0 ≤ T. bn

Let q = (qn )n∈N0 be as at the beginning of the section. Then (bn )n∈N0 = ( is a D-sequence. Further, qn = holds for all n ∈ N0 . Take care, that in [3] qn was defined as

n−1 j=0

qj)

bn+1 bn bn . bn−1

So there is an index shift. However,

in accordance with the construction of q we prefer here to write qn = bbn+1 . n Throughout this section, b = (bn )n∈N0 and q = (qn )n∈N0 will have this meaning. The group of q-adic integers is of importance for us, since q is isomorphic to the character group of Z(b∞ ): Theorem 4.2 [7, (25.2)] The mapping ⎛

q → (Z(b∞ ))∧ , (l j ) j≥0

⎞ n−1  k k → ⎝ + Z → ljbj · + Z⎠ bn b n j=0

is an isomorphism of groups. For the readers’ convenience, we sketch the proof: Proposition 4.3 The mapping : q −→ lim(Z(bn )) = {(kn + bn Z) ∈ Z(bn ) : ∀n ∈ N kn+1 + bn Z = kn + bn Z}, ←  (l j ) j≥0  −→ ( n−1 j=0 l j b j + bn Z)n∈N

is an isomorphism of groups.

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  Proof For every n ∈ N one has nj=0 l j b j − n−1 j=0 l j b j = ln bn ∈ bn Z, which shows that is well-defined. In order to prove that is a homomorphism, it is sufficient to verify that πn ◦ is a homomorphism for every n ∈ N where πn : lim Z(bk ) → Z(bn ) is the canonical ← projection. It is straightforward to show that ◦ : N0 −→ lim(Z(bn )), k = ←

N  j=0

⎛ x j b j  −→ ⎝

n−1 

⎞ xi b j + bn Z⎠ = (k + bn Z)n∈N

j=0

is additive. Since the n-th coordinate of the sum of two elements of q depends only on the first n-coordinates of the summands, it sufficient to consider truncated sequences, in other words, it suffices to show that restricted to 0 = (N0 ) is continuous. But this was shown above. Next we prove that is injective. Therefore, let l = (l j ) j≥0 ∈ ker( ). This means n−1  that l j b j ∈ bn Z for all n ∈ N. For n = 1 we have b1 = q0 divides l0 b0 = l0 . j=0

= 0. Suppose now that l0 = . . . = lm = 0 has Since 0 ≤ l0 < q0 , this yields l0  been shown. Then bm+2 divides m+1 j=0 l j b j = lm+1 bm+1 . Since 0 ≤ lm+1 bm+1 < qm+1 bm+1 = bm+2 , this yields lm+1 = 0. So inductively it is shown that l = (0). In order to prove that is onto, fix (kn + bn Z) ∈ lim← Z(bn ). Since q0 = b1 , we can choose l0 ∈ F q0 such that l0 + b1 Z = k0 + b1 Z. Suppose that l0 , . . . , l m have been chosen so that mj=0 l j b j − km+1 ∈ bm+1 Z. From m and j=0 l j b j + bm+1 Z = km+1 + bm+1 Z km+2 + bm+1 Z = km+1 + bm+1 Z  we see that there exists α ∈ Z such that mj=0 l j b j − km+2 = αbm+1 . Choose lm+1 ∈ Fqm+1 such that lm+1 + qm+1 Z = −α + qm+1 Z. So there exists β ∈ Z such that −α + βqm+1 = lm+1 . This yields m+1 

l j b j − km+2 =

j=0

m 

l j b j − km+2 + lm+1 bm+1 = (α + lm+1 )bm+1 = βqm+1 bm+1 = βbm+2 ∈ bm+2 Z.

j=0

This shows that is onto and completes the proof.



Proposition 4.4 The mapping ∞ ∧

 : lim Z(bn ) −→ Z(b ) , (kn + bn Z) −→ ←

is an isomorphism of groups.



kj k + Z → k · +Z bj bj

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57

Proof Fix (kn + bn Z) ∈ lim← Z(bn ). In order to show that (kn + bn Z) is a well k k defined character of Z(b∞ ) we prove that bkj + Z = b j+l + Z implies k · b jj + Z =  k  k · j+l + Z for all j ∈ N and l ∈ N0 . Assume that l = 0. As k + Z = k + Z implies b j+l

bj

bj

that b j divides k −  k, we obtain that b j divides also (k −  k)k j which is equivalent to kj k j+l  k · b j + Z = k · b j+l + Z. Assume now that for some l ∈ N0 the inductive hypothesis 

k·q

k holds and assume that b j+l+1 + Z = bkj + Z = b j+1j + Z. The inductive hypothesis k k k implies that  k · j+l+1 + Z = k · q j · j+1 + Z = k · j+1 + Z. Since (k j + b j Z) is an b j+l+1

b j+1

bj

element of the projective limit, we have k j+1 − k j ∈ b j Z and hence k ·

k j+1 bj

+Z=

kj bj

+ Z. This shows that  is well-defined. It is straight forward to check that  is a homomorphism. In order to prove that  is onto, fix a character χ ∈ Z(b∞ )∧ . This character χ restricted to the cyclic subgroup { bkn + Z : k ∈ Z} of Z(b∞ ) has the form bkn + Z → kn bkn + Z where kn is unique modulo bn . So χ gives rise to a unique element  (kn + bn Z) ∈ n∈N Z(bn ). From k·

kn 1 qn qn kn+1 kn+1 + Z = χ ( + Z) = χ ( + Z) = +Z= +Z bn bn bn+1 bn+1 bn we conclude that kn+1 − kn ∈ bn Z for all n ∈ N and hence (kn + bn Z) ∈ lim← Z(bn ). Finally, in order to show that  is injective, we fix (kn + bn Z) in the kernel of . This implies that bknn ∈ Z for all n ∈ N or, equivalently, that (kn + bn Z) =  (0 + bn Z). Proof of Theorem 4.2: Combining Propositions 4.3 and 4.4 yields that  ◦ is the desired isomorphism. 

4.2 A Class of Monothetic Reflexive Groups Which do not Have the Weil Property Notation 4.5 For b ∈ D, let τb be the topology of uniform convergence on b on Z, i.e., a neighborhood base at 0 ∈ Z is given by the sets (Um )m∈N where Um = {k ∈ Z :

k + Z ∈ Tm ∀ n ∈ N}. bn

The main result of [3] is: Theorem 4.6 For b ∈ D1 , the character group of (Z, τb ) is algebraically canonically isomorphic to Z(b∞ ). We wish to show that the completion C of (Z, τb ) is a reflexive group.

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1 Theorem 4.7 For b = (bn ) ∈  D the completion C of (Z, τb ) is canonically iso lj  morphic to {(l j ) j∈N0 ∈ q :  q j  −→ 0} where t = min{|t − k| : k ∈ Z}. C is a reflexive group which does not have the Weil-property.

Proof We identify the character group of (Z, τb ) with Z(b∞ ) (Theorem 4.6) and the character group of the discrete group Z(b∞ ) with q (Theorem 4.2). So the second character group of (Z, τb ) can be identified with a subgroup of q . Claim 1: The sequence ( b1n + Z)n∈N0 converges to 0 + Z in the compact open topology in (Z, τb )∧ . Proof of Claim 1: By the definition of the topology of uniform convergence on { b1n + Z : n ∈ N}, the set U1 = { j ∈ Z : bjn + Z ∈ T+ ∀n ∈ N} is a neighborhood of 0 in (Z, τb ). By Facts 2.1 (4), U1 is a compact subset of the character group endowed with the compact-open topology. On the compact set U1 the compact opentopology and the weaker Hausdorff topology σ (Z(b∞ ), Z), the topology of pointwise convergence, coincide. Obviously, b1n + Z ∈ U1 for all n ∈ N0 . Since ( b1n + Z)n∈N0 converges to 0 + Z in the σ (Z(b∞ ), Z)-topology, it also converges to 0 + Z in the compact-open topology.   l  conClaim 2: For every q  η = (l j ) j∈N0 ∈ (Z, τb )∧∧ the sequence  qjj  j≥0

verges to 0 in R. Proof of Claim 2: Since by Claim 1, ( b1n + Z) converges to 0 in (Z, τb )∧ , we  n→∞ bj obtain that η( b1n + Z) = n−1 j=0 l j bn + Z −→ 0 + Z. For n ∈ N, let us put sn =

n−1 

ljbj.

j=0

Then η(

1 sn + Z) = +Z→0+Z bn bn

(1)

and sn−1 ln−1 bn−1 sn−1 ln−1 sn = + = + . bn bn bn bn qn−1 From 0 ≤ l j < q j for all j ∈ N0 , we obtain: sn−1 =

n−2  j=0

and hence

ljbj
n 0 we have η +Z − +Z = lj + Z and bn bn bn j=n 0

n−2  j=n 0

and

lj

bj ln 0 ln−2 1 1 2 1 = + ... + ≤ + ... + ≤ ≤ bn qn−1 qn−2 qn−1 . . . qn 0 qn−1 qn−1 . . . qn 0 +1 qn−1 8m

     ln−1 bn−1   ln−1  1 =     q  ≤ 8m .  b n n−1

Combining the last two inequalities gives η



 n−1 bj 1 k +Z − +Z = lj + bn bn bn j=n 0

Z ∈ Tm . From this we conclude that η − αZ (k) ∈ Um

= {ϑ ∈ (Z, τb )∧∧ : ϑ( b1n + Z) ∈ Tm ∀n ∈ N}. Since the family of sets (Um

)m∈N forms a neighborhood base at 0 in Z∧∧ (Facts 2.1 (5)), we conclude that η ∈ αZ (Z). Claim 4: C is Pontryagin reflexive. Proof of Claim 4: Since C is the completion of the locally quasi-convex metrizable group (Z, τb ), also C is metrizable and locally quasi-convex (Facts 2.1 (2)), in particular, αC is an embedding by Facts 2.1 (3). So it is enough to prove that αC is surjective.

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Let ι : Z → C denote the embedding. It is a folklore fact (and straightforward to prove) that following diagram commutes: Z

ι

C

αZ ∧∧

Z

αC ι∧∧

C

∧∧

This means αC ◦ ι = ι∧∧ ◦ αZ . By Facts 2.1 (1), ι∧ and hence also ι∧∧ are topological isomorphisms. Since αC is an embedding and αC (C) is a closed subgroup of C ∧∧ , we have αC (C) = αC (ι(Z)) = αC (ι(Z)) = ι∧∧ (αZ (Z)) = ι∧∧ (αZ (Z)) = ι∧∧ (Z∧∧ ) = C ∧∧ . This shows that αC is surjective. Claim 5: The group C is does not have the Weil property. Proof of Claim 5: It is sufficient to show that the cyclic subgroup Z is neither precompact nor discrete. Since C is the completion of Z, this is equivalent to: C is neither compact nor discrete. If C were compact, then, by Facts 2.1 (1), C ∧ ∼ = Z∧ = Z(b∞ ) were discrete. But this contradicts Claim 1. If C were discrete, then C ∧ = Z(b∞ ) were compact. But there are no countably infinite compact groups.  The next example presents a group which has the Weil property, but not the strong Weil property and shows that the class of groups having the Weil property is not closed under taking completions.  1 Theorem 4.8 Let b = (bn )n∈N0 ∈ D1 and put β = ∞ n=0 bn ∈ R. (The series converges as bn ≤ 2−n for all n ∈ N0 .) Denote by τ the initial topology on Z2 induced by ϕ : Z2 → (R, τu ) × (Z, τb ), (k, l) → (k + lβ, l) where τu denotes the usual topology on R. Then following assertions hold: 1. (Z2 , τ ) has the Weil property. 2. The completion of (Z2 , τ ) is topologically isomorphic to R × C, where C is the completion of (Z, τb ). 3. The completion of (Z2 , τ ) does not have the Weil property. 4. (Z2 , τ ) does not have the strong Weil property. Proof (1) Let π1 : Z2 → R and π2 : Z2 → Z denote the canonical projections. For (k, l) ∈ Z2 , π1 (k, l) is a cyclic subgroup of R and hence discrete. Since π1 is continuous and injective, we obtain that (k, l) is discrete, in particular, (Z2 , τ ) has the Weil property. (2) We are going to show first that ϕ(Z2 ) is dense in (R, τu ) × (Z, τb ). Fix x ∈ R, m ∈ Z, ε > 0 and N ∈ N. We have to find (k, l) ∈ Z2 such that + Z ∈ Tb N for all n ∈ N. Replacing x by  x = x − mβ |k + lβ − x| < ε and l−m bn and l by l − m, we may assume that m = 0. Further, we may assume that 4b1N < ε. So it remains to be shown that

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61

∀x ∈ R ∀ε > 0 ∀N ∈ N ∃(k, l) ∈ Z2 : |k + lβ − x| < ε and ∀n ∈ N

l + Z ∈ Tb N . (3) bn

Since (bn ) ∈ D1 , it is possible to choose n 0 ∈ N such that bj 1 1 = < qj b j+1 16b N For j ≥ n 0 , put

 l j :=

From

b j+1 1 · bj 8b N

for all j ≥ n 0 .

 and δ j := l j

(4)

bj . b j+1

b j+1 b j+1 1 1 · − 1 < lj ≤ · combined with (4) it follows that: bj 8b N bj 8b N bj bj 1 1 1 ≤ − < lj · = δj ≤ . 16b N 8b N b j+1 b j+1 8b N

This makes it possible to choose n 1 ≥ n 0 such that

n 1 −1

(5)

δ j ≤ x − x
n 1 + 1,

n− j−1

k n1 n1 n1 ∞    bj b j+1 b j+2 1 1 l 1 bn−1 1  1 = lj = δj ... ≤ ≤ ≤ bn bn b j+2 b j+3 bn 8b N 16b N 8b N 16b N 4b N j=n 0

j=n 0

j=n 0

k=1

by Eqs. (4) and (5), so bln + Z ∈ Tb N . We have verified that bln + Z ∈ Tb N holds for all n ∈ N; so Eq. (7), which is also the second statement in Eq. (3), is shown to be correct. In order to check that also first statement in Equation (3) holds, we consider j n1 n1 ∞     bj bj  1 ljbj = lj + lj = lβ = b b b ν=0 ν ν=0 ν j=n 0 j=n 0 j=n 0 ν> j ν    n1 



=:−k∈Z

b bj j  lj + + ... = −k + b j+1 b j+2 j=n 0 n1 n1 n1    1    δj + ljbj = −k + δ j + η. = −k + b j=n 0 j=n 0 ν≥ j+2 ν j=n 0    n1 



(8)

=:η

From Eqs. (4) and (6) we obtain: 0≤η=

n1  j=n 0

≤

n1 

j=n 0

n1   1 k 1 δj ≤ δj ≤ q . . . qν−1 16b N ν≥ j+2 j+1 j=n k≥1



0

2 2 1 1 ε δj · ≤ (δn 1 + x − x) · ≤ (1 + )· < . 16b N 16b N 8b N 8b N 2

For k :=  k + x ∈ Z we have by Eqs. (8) and (5)      n1  ε   |lβ + k − x| = |lβ + k − (x − x)| ≤  δ j − (x − x) + η < δn 1 + ≤ ε. 2  j=n 0  This shows that ϕ(Z2 ) is dense in R × Z. So the completion of (Z2 , τ ) is topologically isomorphic to the completion of (R, τu ) × (Z, τb ), which is R × C. (3) The completion of (Z2 , τ ) does not have the Weil property, since the monothetic group C does not have it.

A Class of Monothetic Reflexive Groups and the Weil Property

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(4) In order to prove that (Z2 , τ ) does not have the strong Weil property, it is sufficient to verify that the finitely generated group (Z2 , τ ) itself is not locally precompact. The completion of (Z2 , τ ) is R × C. The group C is not locally compact, since it does not have the Weil property (Theorem 4.7). Hence also R2 × C is not locally  compact and hence Z2 is not locally precompact. Acknowledgements I wish to express my deep thanks to Elena Martín Peinador for a helpful discussion on this subject. Next to that, I wish to thank the organizers of the “International Meeting on Functional Analysis and Continuous Optimization” for their generous invitation to the highly interesting conference in Elche and for the hospitality I enjoyed there.

References 1. Arhangel’skii, A., Tkachenko, M.: Atlantis Studies in Mathematics. World Scientific Publishing Co., Pte. Ltd., Atlantis Press, Paris, Hackensack, NJ (2008) 2. Außenhofer, L.: Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups. Dissertationes Mathematicae CCCLXXXIV, Warsaw (1999) 3. Außenhofer, L., de la Barrera Mayoral, D.: Linear topologies on Z are not Mackey topologies. J. Pure Appl. Algebra 216(6), 1340–1347 (2012) 4. Außenhofer, L., Dikranjan, D., Giordano Bruno, A.: Topological groups and the Pontryagin-van Kampen duality, de Gruyter Studies in Mathematics, Berlin (2021) 5. Chasco, M.J.: Pontryagin duality for metrizable groups. Arch. Math. (Basel) 70, 22–28 (1998) 6. Gabriyelyan, S.: Groups of quasi-invariance and the Pontryagin duality. Topol. Appl. 157(18), 2786–2802 (2010) 7. Hewitt, E., Ross, K.A.: Abstract harmonic analysis. Structure of topological groups, integration theory, group representations, Grundlehren der Mathematischen Wissenschaften 115, vol. I. Springer, Berlin-New York (1979) 8. Kaplan, S.: Extension of Pontrjagin duality I: infinite products. Duke Math. J. 15, 649–658 (1948) 9. Pestov, V.: Free abelian topological groups and the Pontryagin-van Kampen duality. Bull. Austral. Math. Soc. 52(2), 297–311 (1995) 10. Rolewicz, S.: Some remarks on monothetic groups. Colloq. Math. 13, 27–28 (1964) 11. Smith, M.F.: The Pontrjagin duality theorem in linear spaces. Ann. Math. 56, 248–253 (1952) 12. Tkachenko, M.G.: On completeness of free abelian topological groups, Russian. Acad. Sci. Dokl. Math. 27, 341–345 (1983) 13. Uspenskij, V.V.: Free topological groups on metrizable spaces. Math. USSR Izvestiya 37, 657–679 (1991) 14. Weil, A.: L’intégration dans les groupes topologiques et ses applications (French). Actual. Sci. Ind. no. 869. Hermann et Cie., Paris (1940)

Reciprocation and Pointwise Product in Vector Lattices of Functions Gerald Beer and M. Isabel Garrido

Dedicated to Juan Carlos Ferrando on the occasion of his 65th birthday

Abstract If  is a vector lattice of real-valued functions defined on a set containing the constant functions such that the reciprocal of each nonvanishing member of  remains in , then  is stable under pointwise product. We survey the literature on stability under reciprocation and pointwise product in the context of metric domains, with particular attention given to the uniformly continuous real-valued functions defined on them. For the first time, we present necessary and sufficient conditions for stability for the vector lattice of real-valued coarse maps defined on an arbitrary metric space. Membership of a function to this class means that its associated modulus of continuity is finite-valued, and need not entail continuity of the function. Keywords Vector lattice of real-valued functions · Pointwise product · Reciprocation · Uniformly continuous function · Lipschitz function · Locally Lipschitz function · Coarse map 2010 Mathematics Subject Classification Primary 46E05 · 46E10 · 54C30 · Secondary 46E25 · 26A15 · 54C05 The second author was partially supported by DGES grant PGC2018-097286-B-I00 (Spain) G. Beer (B) Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA 90032, USA e-mail: [email protected] M. Isabel Garrido Instituto de Matemática Interdisciplinar (IMI), Departamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_3

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1 Introduction Let X, d be a metric space with at least two points and let  be a vector lattice of real-valued functions on X . This means for us that  is a vector space over R under pointwise addition of functions and the usual scalar multiplication of functions, and that whenever f, g ∈ , then both f ∨ g := max{ f, g} and f ∧ g := min{ f, g} remain in . Further we make the standard assumption that  contains the constant functions. It is natural to inquire as to what structural assumptions are required on X so that the pointwise product of any two members of  remains in , in which case  becomes a ring with multiplicative identity. Similarly, one can ask under what circumstances the reciprocal of each nonvanishing member of  remains in . When this occurs, we say that  is stable under reciprocation. It is sometimes the case that nothing additional is required on X . Letting C(X, R) denote the continuous real-valued functions on X , the reader can easily verify that this is so in the following cases: • • • •

 = C(X, R) itself;  = { f ∈ C(X, R): f has at most finitely many values};  = { f ∈ C(X, R): f has at most countably many values};  = { f ∈ C(X, R): f is constant in some neighborhood of p} where p ∈ X is fixed; •  = { f ∈ C(X, R) : f ( p1 ) = f ( p2 )} where p1 and p2 are fixed distinct points of X ; •  = { f ∈ C(X, R) : f is constant outside some compact subset of X }. On the other hand, neither the class of uniformly continuous real-valued functions U C(X, R) nor the smaller class of Lipschitz real-valued functions Lip(X, R) is in general stable under pointwise product or reciprocation. Example 1.1 Let [0, ∞) be equipped with the Euclidean metric. In Lip([0, ∞), R), the identity function f (x) := x has nonuniformly continuous square, while g(x) := e−x has nonuniformly continuous reciprocal. It seems a little odd that there is relationship between these stability properties, but indeed there is, as observed by Beer et al. [8]. Before we state the relationship—which in our view is the most important result of this chapter in spite of its simplicity—we make some helpful observations. First, the assumption that a vector space  of real-valued functions be a lattice is equivalent to assuming that f ∈  ⇒ | f | ∈  because f ∨g =

f + g + | f − g| 2

f ∧g =

f + g − | f − g| , 2

while | f | = f ∨ − f. Second, the condition that a vector space  be stable under pointwise product is equivalent to the formally weaker condition f ∈  ⇒ f 2 ∈ , because

Reciprocation and Pointwise Product in Vector Lattices of Functions

fg =

67

1 [( f + g)2 − ( f − g)2 ]. 4

Notice that the next result [8, Theorem 1.1] requires nothing about the domain space or the involved functions topologically speaking. Theorem 1.2 Let  be a vector lattice of real-valued functions on a nonempty set X containing the constant functions. If  is stable under reciprocation, then  is stable under pointwise product. Proof It suffices to show that  is stable under squaring. We first claim that if g ∈  and there exists α > 0 such that for all x ∈ X, |g(x)| = α, then g 2 ∈ . Indeed, note that 1 1 1 − = g 2 − α2 2α(g − α) 2α(g + α) from which g 2 − α2 and then g 2 belong to . Now let f ∈  be arbitrary and put g := | f | + 2 which fulfills the above condition with α = 1. Then g 2 ∈  and f 2 = | f |2 = g 2 − 4| f | − 4. This implies that f 2 ∈  as well.  Corollary 1.3 Let  be a vector lattice of real-valued functions on a nonempty set X containing the constant functions. If  is stable under reciprocation, then whenever f ∈  is non-vanishing and g ∈ , we have gf ∈ . It is remarkable that satisfactory descriptions of those metrics spaces on which the uniformly continuous real-valued functions are stable under pointwise product (resp. reciprocation) were not found until after 2015. The first question was initially resolved by Cabello-Sánchez [18] and the second by Beer et al. [13]. This led to a deluge of related results regarding Lipschitz-type functions, and an increased focus on functions that are strongly locally Lipschitz in various ways. It is the purpose of this chapter to survey the progress over the past years, selectively providing novel proofs in the process. Along the way, we try to convince the reader why certain classes of Lipschitz-type functions are worthy of study and to outline conditions under which the particular functions in the class become fully Lipschitz. We also look at other vector lattices of real-valued functions that until now have fallen through the cracks–most importantly, the real-valued coarse maps. As a warm-up, we look at the class of continuous real-valued functions that have finite limits at infinity. In a metric space X, d where p ∈ X and μ > 0, denote the open ball with center p and radius μ > 0 by Bd ( p, μ). We call a subset E of X bounded provided E is contained in some open ball. Of course this means that diamd (E) := sup{d(x, w) : x, w ∈ E} is finite. We say that a continuous function f : X → R has a finite limit at infinity provided for some α ∈ R and each ε > 0, there exists a bounded set E such that whenever x ∈ X \E, we have | f (x) − α| < ε. In this case, we will write

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limx→∞ f (x) = α. It is easy to verify that the class  of such functions forms a vector lattice of functions containing the constants. To show that they are in general stable under pointwise product, suppose limx→∞ f (x) = α and limx→∞ g(x) = β. Let ε > 0 be arbitrary and choose E 1 , E 2 both bounded such that outside E 1 (resp. E 2 ) we have | f (x) − α| < ε (resp. |g(x) − β| < ε). By the triangle inequality, outside the bounded set E 1 ∪ E 2 , we have | f (x)g(x) − αβ| ≤ | f (x)g(x) − f (x)β + f (x)β − αβ| ≤ (|α| + ε)ε + |β|ε, which can be made arbitrarily small choosing ε small enough. As for stability under reciprocation, the appropriate condition on the metric space is that X, d be bounded. Of course, this condition is sufficient. On the other hand, if the space is unbounded, fix p ∈ X and consider f ∈ C(X, R) defined by  1 if d(x, p) ≤ 1 f (x) = 1 if d(x, p) > 1. d(x, p) While f is nonvanishing and limx→∞ f (x) = 0, its reciprocal does not have finite limit at infinity. As another simple case, consider the bounded continuous real-valued functions on X, d which we denote by Cb (X, R). This is evidently a vector lattice of functions containing the constants that is always stable under pointwise product. As for reciprocation, the desired condition on the domain is that it be a compact metric space. Of course, any continuous function on a compact metric space is bounded, so if f ∈ Cb (X, R) is nonvanishing and X is compact, then 1f belongs to C(X, R) = Cb (X, R). Conversely, if X is not compact, let xn  be a sequence in X without a cluster point. Since the range of the sequence is a closed set and the relative topology on it is discrete, by the Tietze extension theorem [39, p. 103], there exists g ∈ C(X, R) with g(xn ) = n for each n ∈ N. Of course, f ∈ Cb (X, R) defined 1 is nonvanishing, but its reciprocal is not bounded. by f (x) := |g(x)|+1

2 Preliminaries All metric spaces will consist of at least two points. Let X, d be a metric space. We denote the distance from a point x ∈ X to a nonempty subset A of X by d(x, A), where d(x, A) := inf{d(x, a) : a ∈ A}. For ε > 0, the ε-enlargement of a subset A of X is given by Bd (A, ε) := ∪a∈A Bd (a, ε) [6]. When A is nonempty, the following formula is valid: Bd (A, ε) = {x ∈ X : d(x, A) < ε}.

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69

The isolation functional I : X → [0, ∞) [3, 7, 18] for the metric space is defined by the formula I (x) := d(x, X \{x}). Clearly, I (x) = 0 if and only if x is a limit point of X . We denote the set of limit points of the metric space X, d by X . The complement of X consists of the isolated points of X . We call a subset A of X uniformly isolated provided inf{I (a) : a ∈ A} > 0. A sequence xn  in X, d is called Cauchy (resp. cofinally Cauchy) provided given ε > 0, for each pair of integers n and j in some residual (resp. infinite) set of positive integers, we have d(xn , x j ) < ε. The metric space is called complete (resp. cofinally complete) provided each Cauchy (resp. cofinally Cauchy) sequence in it has a cluster point. Cofinally complete metric space have interesting characterizations by covering properties as identified by Rice [36], and also go by the appellation uniformly paracompact metric spaces. While this article focuses on real-valued functions, we prefer to define various classes of functions in the context of functions between arbitrary metric spaces. When the target space is R equipped with the Euclidean metric, then each class we introduce is a vector lattice containing the constants. A function from X, d to a second metric space Y, ρ is continuous if and only if it maps convergent sequences in X to convergent sequences in Y . A stronger requirement is that f map Cauchy sequences in X to Cauchy sequences in Y . This property has been called Cauchy continuity or Cauchy regularity in the literature, and it reduces to ordinary continuity if and only if the domain is a complete metric space [11, 15, 37]. A stronger property still is that f be uniformly continuous, which means that for every ε > 0 there exists δ > 0 such that whenever x, w ∈ X, d(x, w) < δ ⇒ ρ( f (x), f (w)) < ε. The class of metric spaces X, d on which uniform continuity reduces to Cauchy continuity may not be familiar to many readers. Of course, each continuous function on a compact metric space with values in a second metric space is uniformly continuous. But this property is not characteristic of compactness of the domain. For example, this is so if the domain is N equipped with the Euclidean metric. A metric space X, d is called a UC-space provided each continuous function on it with values in a second metric space is uniformly continuous. Visually speaking, a metric space is most easily seen to be a UC-space if / Bd (X , ε) ⇒ I (x) > δ. and only if X is compact, and ∀ε > 0 ∃δ > 0 such that x ∈ In simple words, the mathematical statement says that if we stay away from the set of limit points by some positive distance, what remains is a uniformly isolated set. In fact, a metrizable topological space admits a compatible UC-metric if and only if its set of limit points is compact [5, 35]. The class of UC-spaces sits strictly between the compact metric spaces and the complete metric spaces. Here is a list of other characteristic properties of a UC-space X, d, all established prior to 1960, which is by no means exhaustive (see, e.g., [3, 6, 28, 34]). • C(X, R) = U C(X, R); • For each pair of disjoint nonempty closed subsets A and B of X , we have inf{d(a, b) : a ∈ A, b ∈ B} > 0; • each open cover of X has a Lebesgue number;

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• whenever xn  is a sequence in X with limn→∞ I (xn ) = 0, then xn  clusters. We are now ready to state necessary and sufficient conditions on X, d for uniform continuity to reduce to Cauchy continuity for functions defined on X : the completion of X, d is a UC-space [4, 26]. A nontrivial example of such a space is N ∪ { n1 : n ∈ N} equipped with the usual Euclidean metric. Returning to the main line of discussion, uniform continuity is implied by Lipschitz continuity. A function f : X, d → Y, ρ is called λ-Lipschitz if there exists λ ≥ 0 such that whenever x, w ∈ X, we have ρ( f (x), f (w)) ≤ λd(x, w). A 1Lipschitz function is often called nonexpansive. We call f : X → Y locally Lipschitz provided its restriction to some neighborhood of each point in X is Lipschitz. While a locally Lipschitz function must be continuous, it need not be Cauchy continuous (consider f (x) = x1 defined on (−∞, 0) ∪ (0, ∞)). Various subclasses of the locally Lipschitz functions that have been intensely studied over the last 10 years will be indicated in the sequel. By a bornology B on a metric space X, d we mean a family of subsets of X that contains the singletons, that is hereditary, and that is stable under finite unions. The smallest bornology is the family of its finite subsets and the largest is the power set of X . The term is derived from the French, as borné means bounded in that language. Indeed, the bounded subsets of X form a bornology. Within the bounded subsets we indicate some other basic bornologies, listed in increasing size. First comes the relatively compact subsets, that is, the family of subsets with compact closure. Next on the list comes the totally bounded subsets, i.e., those subsets that are contained for each ε > 0 in the ε-enlargement of some finite set. Larger still is the bornology of Bourbaki bounded subsets, also called the finitely chainable subsets. By an ε-chain of length n from a ∈ X to b ∈ X , we mean a finite sequence of points x0 , x1 , x2 , . . . , xn in X such that x0 = a, xn = b, and for each j ∈ {1, 2, 3, . . . , n}, we have d(x j−1 , x j ) < ε. Note that repeats are not precluded in our definition. With this notion in mind, A ⊆ X is declared Bourbaki bounded [3, 9, 16, 18, 22] provided ∀ε > 0 there exists a finite subset F of X and n ∈ N such that each point of A can be joined to some point of F by an ε-chain of length n. We remark that when X = Rn equipped with the Euclidean metric, all of these bornologies are the same because each bounded set is relatively compact. Considerable energy has been given to characterizing these bornologies (1) in terms of subsequences that each sequence in a member of the bornology must have; (2) as bornologies on which each member of a certain class of continuous functions on X is bounded. With respect to (1), we only mention this classical fact [39, p. 182]: A ⊆ X is totally bounded if and only if each sequence in A has a Cauchy subsequence. From this equivalence, the reader is invited to show that A is totally bounded if and only if each sequence in A is cofinally Cauchy. We now list the most important results relative to (2): • A ⊆ X is relatively compact iff each continuous function on X is bounded on A; • A ⊆ X is totally bounded iff each Cauchy continuous function on X is bounded on A [9];

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• A ⊆ X is Bourbaki bounded iff each uniformly continuous function on X is bounded on A [9, 31]; • A ⊆ X is bounded iff each Lipschitz function on X is bounded on A. Membership to each bornology is actually determined by the real-valued functions of the type specified. Thus, for example, a subset A of a metric space is bounded if and only if each real-valued Lipschitz function on X is bounded when restricted to A (if A is unbounded, consider the nonexpansive map x → d(x, p) where p ∈ X is arbitrary). In view of the discussion in the last paragraph of our introductory section, the reader should be able to easily prove the next result. Theorem 2.1 Let B be a bornology on a metric space X, d. Let  = { f ∈ C(X, R) : f is bounded on each member of B}. Then  is a vector lattice of functions containing the constants that is always stable under pointwise products. Further,  is stable under reciprocation if and only if each member of B is relatively compact.

3 Cauchy Continuity and Uniform Continuity Stability under pointwise product and stability under reciprocation for the class of Cauchy continuous real-valued functions requires no imagination to resolve. Theorem 3.1 Let X, d be a metric space and let  be the vector lattice of Cauchy continuous real-valued functions. Then  is always stable under pointwise product, and  is stable under the reciprocation if and only if X, d is a complete metric space. Proof It is a standard exercise in an introductory analysis course (using the boundedness of each Cauchy sequence) to show that the product of two real-valued Cauchy sequences is again Cauchy. Stability under pointwise product follows from this. Turning to reciprocation, if X is complete, then the class of real-valued Cauchy continuous functions agrees with C(X, R) and hence is stable under reciprocation. If ˆ The completion is a comX, d is not complete, denote its completion by  Xˆ , d. plete metric space and X sits isometrically as a dense subset of Xˆ . Take pˆ ∈ Xˆ \X . ˆ p) Then d(·, ˆ is a uniformly continuous (in fact 1-Lipschitz) function on the completion; its restriction to X is a nonvanishing uniformly continuous function, but its reciprocal is not uniformly continuous because pˆ can be approached by a sequence ˆ n , p) ˆ n+1 , p), ˆ ≥ 2d(x ˆ in X . More precisely, choosing for each n ∈ N xn ∈ X with d(x we compute 1 ˆ n+1 , p) ˆ d(x

−

1 ˆ n , p) ˆ d(x

≥

ˆ n+1 , p) ˆ d(x 1 = . ˆ n , p) ˆ n+1 , p) ˆ n , p) ˆ · d(x ˆ ˆ d(x d(x

As a result, the reciprocal maps the Cauchy sequence xn  to an unbounded real sequence, so the reciprocal fails to be Cauchy continuous. 

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In the case of U C(X, R), stability under reciprocation is more transparent than stability under pointwise product, and we treat that first. The proof of necessity we give here bears no resemblance to the one given by Beer, Garrido and Meroño in [13, Theorem 2.2]. Happily, our new proof is much more elementary. We use the fact that in a metric space that is not UC, we can find disjoint nonempty closed subsets that are asymptotic. Theorem 3.2 The vector lattice U C(X, R) for a metric space X, d is stable under reciprocation if and only X, d is a UC-space. Proof If X, d is a UC-space, then by definition U C(X, R) = C(X, R) and C(X, R) is always stable under reciprocation. Conversely, suppose X, d is not a UC-space. As pointed out earlier, this means that for some pair of disjoint nonempty closed subsets A and B, we have inf{d(a, b) : a ∈ A, b ∈ B} = 0. We will use this equivalent nonsymmetric condition: inf{d(a, B) : a ∈ A} = 0. So for each n ∈ N pick an ∈ A with 0 < d(an , B) < n1 . For each n choose bn ∈ B such that d(an , bn ) < 23 d(an , B). We obtain this inequality string: (∗)

1 1 1 d(bn , A) ≤ d(bn , an ) < d(an , B). 3 3 2

Let f ∈ Lip(X, R) be given by f (x) := 13 d(x, A) + d(x, B). Of course, f is uniformly continuous and nonvanishing by the disjointness of the closed sets A and B. Note that limn→∞ 1f (an ) = ∞ because f (an ) < n1 . Using condition (∗), we compute 1 1 2 1 (bn ) = 1 > 1 = 2 · (an ). f f d(bn , A) + d(bn , B) d(an , A) + d(an , B) 3 3 We conclude that 1f fails to be uniformly continuous because limn→∞ d(bn , an ) = 0  while limn→∞ 1f (bn ) − 1f (an ) = ∞. We now turn to pointwise product where the initial advance was made by CabelloSánchez [18, Theorem 3.5]. We call X, d a Cabello-Sánchez space provided each subset of X is either Bourbaki bounded or contains an infinite uniformly isolated subset. A key tool in our analysis is the McShane extension theorem: let A be a nonempty subset of X and let f be a bounded uniformly continuous real-valued function on A. If the range of f is contained in the closed interval [α, β], then f has an extension to U C(X, R) whose range is also contained in [α, β] [24, 32]. Statement (2) of the next theorem first appears in [13]. Theorem 3.3 Let X, d be a metric space. The following conditions are equivalent: (1) X, d is a Cabello-Sánchez space;

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(2) whenever xn  is a sequence in X with limn→∞ I (xn ) = 0, then {xn : n ∈ N} is Bourbaki bounded; (3) U C(X, R) is stable under pointwise product; (4) whenever f ∈ U C(X, R) and h ∈ U C(X, R) is bounded, then f h is uniformly continuous. Proof (1) ⇒ (2). If xn  is a sequence in X along which the isolation functional tends to zero, then if A is an infinite subset of {xn : n ∈ N}, we must have inf {I (a) : a ∈ A} = 0, i.e., A is not uniformly isolated. Statement (1) guarantees that {xn : n ∈ N} is Bourbaki bounded. (2) ⇒ (3). Suppose statement (2) holds while statement (3) fails. We can find f, g ∈ U C(X, R) such that f g ∈ / U C(X, R). There exists ε > 0 and for each n ∈ N points xn , wn ∈ X with 0 < d(xn , wn ) < n1 but (♣) | f (xn )g(xn ) − f (wn )g(wn )| ≥ ε. Evidently, both limn→∞ I (xn ) = 0 and limn→∞ I (wn ) = 0, so that by (2) A := {x1 , w1 , x2 , w2 , . . .} is Bourbaki bounded and as a result, both f and g are bounded on A. We conclude that the pointwise product is uniformly continuous restricted to A, and this produces a contradiction to (♣). (3) ⇒ (4). This is trivial. (4) ⇒ (1). We repeat the beautiful argument of [18, Theorem 3.5]. If (1) fails, there exists A ⊆ X that is neither Bourbaki bounded nor contains an infinite uniformly isolated subset. The latter condition means ∀α > 0, {a ∈ A : I (a) ≥ α} is finite. The first statement guarantees the existence of f ∈ U C(X, R) such that f | A is unbounded. We can assume without loss of generality that f has nonnegative values. As such, we can choose a sequence xn  in A such that for each n ∈ N, f (xn+1 ) ≥ f (xn ) + 1 ≥ n + 1. By uniform continuity, there exists δ > 0 such that n = m ⇒ d(xn , xm ) ≥ δ. Since limn→∞ I (xn ) = 0, by passing to a subsequence, we can find a second sequence wn  in X such that for each n, 0 < d(xn , wn ) < 13 δ and with limn→∞ d(xn , wn ) = 0. Since the set of terms S of the spliced sequence x1 , w1 , x2 , w2 , . . . are distinct, the function h 0 : S → [0, 1] given by h 0 (xn ) = n1 and h 0 (wn ) = 0 is well-defined and uniformly continuous. By McShane’s theorem, we may extend h 0 to h ∈ U C(X, [0, 1]). Since for each n we have f h(xn ) ≥ 1 while  f h(wn ) = 0, the pointwise product f h fails to be uniformly continuous. Recall that X, d is a UC-space if and only if each sequence in it along which the isolation functional goes to zero necessarily clusters. This can be restated as follows: X, d is a UC-space if and only if whenever xn  is a sequence in X, d with limn→∞ I (xn ) = 0, then {xn : n ∈ N} is relatively compact. In view of statement (2), the Cabello-Sánchez spaces contain the UC-spaces (but this is to be anticipated from Theorem 1.2). The intermediate condition: whenever xn  is a sequence in X, d with limn→∞ I (xn ) = 0, then {xn : n ∈ N} is totally bounded amounts to saying that whenever xn  is a sequence in X, d with limn→∞ I (xn ) = 0, then xn  has a Cauchy subsequence. This occurs if and only if X, d has UC-completion. Again, these are the metric spaces on which Cauchy continuity implies uniform continuity [4, 26]. A. Bouziad and E. Sukhacheva [16], who chose to work more generally in the framework of uniform spaces, subsequently compiled a comprehensive list of equiv-

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alent conditions for when U C(X, R) is a ring. In this setting, the Katetov extension theorem [27] must be used instead of the McShane theorem. Some are very attractive function space conditions. For a purely metric approach to some of these equivalences, the reader may consult [7]. In the following result, we record some of their more accessible conditions - both internal and function space in nature—all written in metric space language. Theorem 3.4 The following conditions are equivalent for a metric space X, d: (1) U C(X, R) is stable under pointwise product; (2) for each f ∈ U C(X, R) there exists μ > 0 such that {x ∈ X : | f (x)| > μ} is uniformly isolated; (3) for each f ∈ U C(X, R) there exists μ > 0 such that f is uniformly locally constant on {x ∈ X : | f (x)| > μ}; (4) for each f ∈ U C(X, R) and for each g ∈ C(R, R), g ◦ f is uniformly continuous; (5) whenever A ⊆ X is not Bourbaki bounded, there exist ε > 0 and an infinite subset E of A such that Bd (E, ε) ⊆ A; (6) whenever (xn , wn ) is a sequence in X × X such that for each n, xn = wn , and limn→∞ d(xn , wn ) = 0, then {x1 , w1 , x2 , w2 , . . .} is Bourbaki bounded. Bouziad and Sukhacheva raise the following natural question that, to the best of our knowledge, has not been settled: exactly when does a metrizable space admit a compatible metric such that U C(X, R) is stable under pointwise product? We start with a class of metrizable spaces such that the uniformly continuous real-valued functions on each member fail to form a ring no matter what compatible metric is used. We will henceforth call a subset of a metrizable space nontrivial if it contains at least two points. Example 3.5 Let X, τ  be a metrizable space that is a disjoint union of uncountably many (connected) components, all of which are nontrivial compact subsets of X , and such that the union of any subfamily of the family of components C is a closed subset of X . An example of such a space is R × [0, 1] equipped with the topology produced by the metric  1 if α1 = α2 ρ((α1 , β1 ), (α2 , β2 )) := |β1 − β2 | if α1 = α2 . Now if d were a compatible metric for such a metrizable space X, τ  that made U C(X, R) a ring, then according to the Cabello-Sánchez criterion, X must be Bourbaki bounded subset because X = X . We show this leads to a contradiction. For each n ∈ N, choose a finite subset Fn of X and kn ∈ N so that each point of X can be joined to some point of Fn by a n1 -chain of length kn . Let En be the finite family of components that hit Fn . As ∪∞ n=1 En is a countable family of components, there must be a component C0 that is not so enumerated. By the compactness of

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C0 and the closedness of X \C0 = ∪{C ∈ C : C = C0 }, we can find ε > 0 such that Bd (C0 , ε) = C0 . Choose n with n1 < ε and then c0 ∈ C0 and p ∈ Fn so that p can be joined to c0 by a n1 -chain of length kn , say {x0 , x1 , . . . , xkn }. Now there is a smallest j ∈ {1, 2, . . . , kn } such that x j ∈ C0 . Then d(x j−1 , x j ) < ε, and as x j−1 ∈ / C0 , this violates Bd (C0 , ε) = C0 . We remark that the concrete representative of the class of metrizable spaces that we presented in the last example can be viewed as the topological product of R equipped with the discrete topology and [0,1] equipped with its usual topology. Thus, the topological product of two metric spaces such that the uniformly continuous realvalued functions on each form a ring (for R use the zero-one metric and for [0,1] use any compatible metric) need not be a Cabello-Sánchez space with respect to any compatible metric for the product topology! Bouziad and Sukhacheva [16, Corollary 3.12] prove that if X, τ  has a compatible metric d such that X becomes Bourbaki bounded in itself, that is, if the metric subspace X , d is Bourbaki bounded, then there is a uniformly equivalent metric on X (as introduced in [5]) with respect to which U C(X, R) is stable under pointwise product. We do not know if their condition is necessary, while of course Bourbaki boundedness of X as a subset of X is necessary. As a consequence of their result, a class of metrizable spaces having a compatible metric for which U C(X, R) is a ring are those for which X is separable in its relative topology. If X is empty, then the zero-one metric on X does the job. Otherwise, by the Urysohn metrization theorem [39, p. 166], X can be given a compatible totally bounded metric, which can be extended to a compatible metric for the entire space by Hausdorff’s extension theorem [23] because X is a closed subset of X . On the other hand, separability of X is certainly not necessary: take for X the closed unit ball in any nonseparable Banach space equipped with its relative topology.

4 Lipschitz-Type Functions The following results are taken from [8]. The first is a folk theorem, and we give a new proof of the second. Theorem 4.1 Let X, d be a metric space. Then Lip(X, R) is stable under pointwise product if and only if the space is bounded. Proof If X, d is bounded, then each element of Lip(X, R) is bounded and so the function space is stable under pointwise product. Conversely, if X, d is unbounded, let p ∈ X be arbitrary. While f (x) = d(x, p) is 1-Lipschitz, its square is not Lipschitz. 

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To deal with reciprocation, we will use this classical result [24, Theorem 6.2]: Let A be a nonempty subset of X, d and let f : A → R be λ-Lipschitz. Then f has a λ-Lipschitz extension g to X . A formula for such an extension g is the following: g(x) := infa∈A ( f (a) + λd(x, a)). If f is nonnegative, then g so described is nonnegative as well. If A is closed and f is positive-valued, then replacing g by g + d(·, A) gives a strictly positive Lipschitz extension (with a larger Lipschitz constant). We also note that this Lipschitz extension theorem can be used to derive the McShane theorem under a uniformly equivalent remetrization [7]. Theorem 4.2 Let X, d be a metric space. Then Lip(X, R) is stable under reciprocation if and only if the space is compact. Proof Suppose X, d is compact and f : X → R is λ-Lipschitz and nonvanishing. Let α denote the minimum value of | f | on X . Then for each x, w ∈ X , we obtain |

1 | f (w) − f (x)| λ 1 − |≤ ≤ 2 · d(x, w). 2 f (x) f (w) α α

For the converse, we show that X, d is complete and totally bounded provided it ˆ is stable under reciprocation (see, e.g., [39, p. 182]). If completeness fails, let  Xˆ , d ˆ p) be the completion. Taking pˆ ∈ Xˆ \X , the restriction of d(·, ˆ to X is 1-Lipschitz and nonvanishing, but its reciprocal is not Lipschitz because X is dense in Xˆ . We know from Theorems 1.2 and 4.1 that X, d is at least bounded. If it fails to be totally bounded, we can find δ > 0 and a sequence xn  in X such that n = j ⇒ d(xn , x j ) ≥ δ. Let E be the set of terms of the sequence. Define f : E → R by  1 if n = 1 f (xn ) = . δ otherwise nd(x1 ,xn ) It is routine to verify that f is 1δ -Lipschitz. Extend f to a positive-valued Lipschitz function h on X . Finally, h1 fails to be Lipschitz, because for n > 1, we have 1 1 (xn ) = (xn ) ≥ n. h f This is impossible, as a Lipschitz function defined on a bounded metric space must be bounded.  The following omnibus theorem has many applications. Theorem 4.3 Let A be a family of nonempty subsets of X, d and let  be the family of continuous real-valued functions that are Lipschitz when restricted to each member of A.

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(1) if each member of A is bounded, then  is stable under pointwise product; (2)  is stable under reciprocation if and only if each member of A is relatively compact. Proof Statement (1) is obvious because each Lipschitz function on a bounded set is bounded, and the product of two bounded Lipschitz functions is Lipschitz. For sufficiency in (2) suppose f ∈  is nonvanishing. Let A ∈ A; by assumption there exists λ > 0 such that f | A is λ-Lipschitz. By continuity, f |cl(A) is also λLipschitz, so by Theorem 4.2, 1f is Lipschitz on cl(A) and thus on A. For necessity, suppose A ∈ A is not relatively compact. The proof of Theorem 4.2 shows that there is a Lipschitz function f : cl(A) → (0, ∞) whose reciprocal is not Lipschitz on cl(A) and therefore is not Lipschitz on A by continuity. But since cl(A) is closed, f has a positive Lipschitz-constant preserving extension g to X and of course, g ∈ , because the restriction of any Lipschitz function to any nonempty subset will be  Lipschitz. But g1 cannot be in  as its restriction to A agrees with 1f . Corollary 4.4 Let X, d be a metric space, and let  be the vector lattice of functions that are Lipschitz when restricted to each bounded subset of X . Then  is always stable under pointwise product and is stable under reciprocation if and only if each closed and bounded subset of X is compact. Metric spaces in which closed and bounded sets are compact are simply those for which each bounded subset is relatively compact, because the closure of each bounded subset is also bounded. They are usually called boundedly compact in the literature. By a classical result of Vaughan [38], a metrizable topological space has a compatible boundedly compact metric if and only if it is locally compact and separable. Corollary 4.5 Let X, d be a metric space, and let  be the vector lattice of functions that are Lipschitz when restricted to each relatively compact subset. Then  is always stable under pointwise product and reciprocation. It is well-known that the class of functions between metric spaces that are Lipschitz when restricted to each relatively compact subset agrees with the class of locally Lipschitz functions, i.e., those functions that are Lipschitz on some neighborhood of each point of the domain (see, e.g., [10, Theorem 4.2]). An important fact about them is that they are uniformly dense in C(X, R) for any metric space X, d [11, 19, 21, 33]. Uniform density follows easily from this deep result of Frolik [20]: given an open cover V of a metric space X, d, there exists a locally finite partition of unity { pi : i ∈ I } consisting of Lipschitz functions on X subordinated to the cover (see also [19]). Corollary 4.6 Let X, d be a metric space, and let  be the vector lattice of functions that are Lipschitz when restricted to each totally bounded subset. Then  is always stable under pointwise product, and is stable under reciprocation if and only if d is a complete metric.

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Proof A subset of a metric space is compact if and only if it is both totally bounded and complete (viewed as a metric space equipped with the metric d). If A is totally bounded, then so is its closure. Now if d is complete then so is each closed metric subspace, and thus cl(A) is complete and totally bounded. Thus, with completeness of d, each totally bounded subset is relatively compact. Conversely, if each totally bounded set is relatively compact, then in particular the range of each Cauchy sequence in X is relatively compact. Easily, each Cauchy sequence must then have a cluster point.  The class of functions between metric spaces that are Lipschitz when restricted to each totally bounded subset was introduced by Beer and Garrido [11]. A function has this property if and only if its restriction to the range of each Cauchy sequence in X is Lipschitz [11, Proposition 3.4]. For this reason, the authors called such a function Cauchy-Lipschitz. Evidently each such function is Cauchy continuous. The authors proved that the Cauchy-Lipschitz real-valued functions are uniformly dense in CC(X, R). Corollary 4.7 Let X, d be a metric space, and let  be the vector lattice of functions that are Lipschitz when restricted to each Bourbaki bounded subset. Then  is always stable under pointwise product, and is stable under reciprocation if and only if each Bourbaki bounded subset of X is relatively compact. Garrido and Meroño [22] characterized the relative compactness of each Bourbaki bounded subset sequentially. Call a sequence xn  in a metric space X, d BourbakiCauchy provided for each ε > 0, there exist {m, n 0 } ⊆ N such that ∀n ≥ n 0 , ∀ j ≥ n 0 there exist an ε-chain of length m from xn to x j . Then each Bourbaki bounded subset of X is relatively compact if and only if each Bourbaki-Cauchy sequence in X clusters. This works since a subset of X is Bourbaki bounded if and only is each sequence in it has a Bourbaki-Cauchy subsequence. But there are important classes of locally Lipschitz functions that have not been characterized as those functions that are Lipschitz when restricted to each member of some family of subsets of the domain. We call f : X, d → Y, ρ Lipschitz in the small if there exist δ > 0 and λ > 0 such that d(x, w) < δ ⇒ ρ( f (x), f (w)) ≤ λd(x, w). This class was introduced by Luukkainen [30]. Each Lipschitz in the small function is obviously uniformly continuous. If N is equipped with the Euclidean metric, the functions f : N → R defined by f (n) = n 2 is a Lipschitz in the small function (take δ = 21 and λ = 1), but it is not Lipschitz. The reader is invited to check that each bounded Lipschitz in the small function must be Lipschitz [21]. Paralleling our two uniform density results already mentioned, for any metric space X, d, the Lipschitz in the small real-valued functions on X are uniformly dense in U C(X, R) [10, 21]. The family of subsets of X on which each Lipschitz in the small function is actually Lipschitz has been called the family of small determined subsets. Membership to this family is determined by real-valued Lipschitz in the small functions, and such subsets need not be stable under finite union. It is also the case that a function that is Lipschitz

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restricted to each small determined set need not be Lipschitz in the small. The family of small determined subsets has been characterized in a technical way by Leung and Tan [29] The next result again comes from [8] and parallels our results for uniformly continuous real-valued functions. Theorem 4.8 Let X, d be a metric space. Then the real-valued Lipschitz in the small functions are stable under reciprocation if and only if X, d is a UC-space. They are stable under pointwise product if and only if X, d is a Cabello-Sánchez space. It is a classical fact that for any metric space X, d, the class of bounded realvalued Lipschitz functions Lipb (X, R) is uniformly dense in the bounded uniformly continuous real-valued functions U Cb (X, R) [24, Theorem 6.8]. One wonders if better uniform approximations can be obtained for elements of the bounded real-valued continuous functions or the bounded real-valued Cauchy-continuous functions. So for example, one might look for a nice subclass of the locally Lipschitz real-valued functions whose bounded members are uniformly dense in Cb (X, R). A Lipschitz in the small function is one that is locally Lipschitz at each point, and where a ball of common size on which the function is Lipschitz and a common Lipschitz constant can be chosen independent of the point. If it is only possible to choose a ball of common size on which the function is Lipschitz, then the function is called uniformly locally Lipschitz [9]. Each such function is Cauchy-Lipschitz. As an example, by the mean value theorem, each continuously differentiable function f : R → R has this property, because its derivative is bounded on each ball of radius 1. It would be interesting to identify the uniform closure of the real-valued uniformly locally Lipschitz functions within CC(X, R). It is known, however, that these functions will be uniformly dense in CC(X, R) if and only if the completion of X, d is cofinally complete [10]. Again from [8] comes the next result. Theorem 4.9 Let X, d be a metric space. Then the real-valued uniformly locally Lipschitz functions on X are always stable under pointwise product, and are stable under reciprocation if and only if X, d is cofinally complete. On the other hand, a function that is locally Lipschitz and for which a common locally Lipschitz constant exists for some neighborhood of each point has been called locally equi-Lipschitz by Aggarwal and Cobza¸s [1]. This class was considered superficially under a different name by Heinonen much earlier [25]. Little progress has been made as to when this class is stable under pointwise product. But the authors were able to dispense with stability under reciprocation. Theorem 4.10 Let X, d be a metric space. Then the real-valued locally equiLipschitz functions are stable under reciprocation if and only if the set of limit points of X is compact.

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Proof We give a new proof of sufficiency, and refer the reader to [1] for necessity which is much less transparent. Let f : X → R be locally equi-Lipschitz and nonvanishing. Since 1f is Lipschitz with respect to any Lipschitz constant you like in some neighborhood of each isolated point, we need only show that there is a common local Lipschitz constant for the reciprocal at each limit point of X . By the continuity

of f and the compactness of X , α := min{| f (x)| : x ∈ X } is positive. Let λ > 0 be a common local Lipschitz constant for f . Let p be an arbitrary limit point of X . Choose δ > 0 such that f is λ-Lipschitz restricted to Bd ( p, δ) and ∀x ∈ Bd ( p, δ) we have | f (x)| ≥ α2 . Then if {x, w} ⊆ Bd ( p, δ), we compute 1 | f (w) − f (x)| 4λ · d(x, w) 1 ≤ . | (x) − (w)| = f f | f (x)|| f (w)| α2 These estimates show that reciprocal.

4λ α2

is a common local Lipschitz function for the 

It follows from our last result that compactness of X is sufficient for the locally equi-Lipschitz real-valued functions to be stable under pointwise product. The mathematical community awaits important applications of the class of locally equi-Lipschitz functions.

5 Real-Valued Coarse Maps Although we were partially guided by the approach of Cabello-Sánchez [18] in this section, our results here are simply not anticipated by anything in the literature. Let X, d and Y, ρ be metric spaces. If f : X → Y , the modulus of continuity of f [2, 7, 14, 27, 32] is the function ω f on [0, ∞) defined by the formula ω f (t) := sup{ρ( f (x), f (w)) : d(x, w) ≤ t}. In our opinion, and despite its wide-spread use, this language is an egregious misnomer, as ω f is really a growth modulus for the function. Clearly, ω f (0) = 0 and ω f is an increasing non-negative extended real-valued function. Uniform continuity of f is equivalent to the continuity of ω f at the origin, while f is λ-Lipschitz if and only if ω f (t) ≤ λt for all t > 0. On the other hand, f is Lipschitz in the small provided for some λ > 0 and δ > 0, we have ω f (t) ≤ λt for all t ∈ [0, δ). It is not hard to show that a Lipschitz in the small function whose modulus of continuity has an affine majorant is already Lipschitz. McShane [32, Theorem 2] showed that if f is real-valued and uniformly continuous on some subset A of X and its modulus of continuity relative to A has an affine majorant, then f has a uniformly continuous extension to the entire space.

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A function f is called a coarse map provided ω f (t) < ∞ for all nonnegative t. Coarseness is a uniform constraint on the rate of growth of a function, stronger than simply requiring that the function map bounded subsets of the domain to bounded subsets of the target space. A function can be nowhere continuous yet still be a coarse map, e.g., the Dirichlet function on R. The verification of this alternative description of coarseness is left to the reader. Proposition 5.1 Let X, d and Y, ρ be metric spaces. A function f : X → Y is a coarse map if and only if for each t > 0 there exists Mt > 0 such that whenever A ⊆ X is nonempty and diamd (A) ≤ t, then diamρ f (A) ≤ Mt . Beer and Garrido [12] called f : X → Y bounded in the small with distance control t if ω f (t) < ∞ for some positive t. In this case, limt→0+ ω f (t) is a finite nonnegative number. By definition, a function between metric spaces is a coarse map provided it is bounded in the small with distance control t for all positive t. In the spirit of the Garrido-Jaramillo theorem [21] characterizing the Lipschitz property for functions between metric spaces in terms of following compositions, Beer and Garrido [6, Proposition 4.3] showed that f : X → Y is bounded in the small with distance control t if and only if ∀g ∈ Lip(Y, R), g ◦ f is bounded in the small with distance control t. It follows that f is a coarse map if and only if ∀g ∈ Lip(Y, R), g ◦ f is a coarse map. It is an easy exercise to show that the real-valued coarse maps defined on a metric space X, d form a vector lattice containing the constants. Obviously, the squaring function f (x) = x 2 on R is a product of coarse maps which is not itself a coarse map. In fact, ω f (t) = ∞ whenever t > 0. Our next result gives necessary and sufficient conditions on the structure of a metric space for the coarse maps to be stable under pointwise product. These conditions remain necessary if we confine our attention to the continuous coarse maps. Notice how they resemble the CabelloSánchez conditions relative to U C(X, R) in spirit. Theorem 5.2 Let X, d be a metric space. The following conditions are equivalent: (1) each subset A of X is either bounded or satisfies the condition sup{I (a) : a ∈ A} = ∞; (2) the pointwise product of any two real-valued coarse maps on X remains a coarse map; (3) the pointwise product of any two real-valued continuous coarse maps on X remains a coarse map. Proof (1) ⇒ (2). Suppose condition (2) fails. Let f and g be real-valued coarse maps such that for some t0 > 0 we have ω f g (t0 ) = ∞. For each n ∈ N we can find xn and wn in X such that d(xn , wn ) ≤ t0 while | f (xn )g(xn ) − f (wn )g(wn )| ≥ n. Let A := {xn : n ∈ N} ∪ {wn : n ∈ N}. By construction, sup{I (a) : a ∈ A} ≤ t0 . Next, suppose A is bounded to reach a contradiction. We can find t1 > t0 such that diamd (A) ≤ t1 . Since ω f (t1 ) and ωg (t1 ) are both finite, we can find M f > 0 (resp. Mg > 0) such that for all a ∈ A we have both | f (a)| ≤ M f and |g(a)| ≤ Mg . We compute for each n ∈ N

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| f (xn )g(xn ) − f (wn )g(wn )| ≤ | f (xn )||g(xn ) − g(wn )| + |g(wn )|| f (xn ) − f (wn )| ≤ M f ωg (t0 ) + Mg ω f (t0 ), and this contradicts supn∈N | f (xn )g(xn ) − f (wn )g(wn )| = ∞. Thus, condition (1) fails if (2) fails. (2) ⇒ (3). This is trivial. (3) ⇒ (1). Suppose (1) fails, i.e., there is a subset A of X that is unbounded and sup{I (a) : a ∈ A} < ∞. We intend to produce two continuous real-valued coarse maps whose pointwise product is not a coarse map. One of these functions will in fact be Lipschitz and the other bounded. Fix a1 ∈ A and choose inductively a2 , a3 , a4 , . . . in A such that d(an+1 , a1 ) ≥ d(an , a1 ) + 2n for each positive integer n. Whenever j > n ≥ 1, we have ♠

d(a j , an ) ≥ d(a j , a1 ) − d(an , a1 ) ≥ d(an+1 , a1 ) − d(an , a1 ) ≥ 2n .

Choose n ≥ 3 so large that sup{I (a) : a ∈ A} < 2n−2 . For each j ≥ n, we can find x j ∈ X such that 0 < d(a j , x j ) ≤ 2n−2 . It follows from ♠ that for all j ≥ n, d(a j , {ak : k = j and k ≥ n} ∪ {xk : k = j and k ≥ n}) ≥ 1, and d(x j , {ak : k = j and k ≥ n} ∪ {xk : k = j and k ≥ n}) ≥ 1. It follows that {a j : j ≥ n} and {x j : j ≥ n} are disjoint closed sets. By Urysohn’s lemma, there is a continuous function f : X → [0, 1] such that for all j ≥ n we have f (a j ) = 1 and f (x j ) = 0. Of course, f is a coarse map because it is a bounded function. Now let g := d(a1 , ·), a 1-Lipschitz function, and therefore a coarse map as well. Since | f (a j )g(a j ) − f (x j )g(x j )| = g(a j ) = d(a1 , a j ) can be made arbitrarily large for j ≥ n, we have ω f g (2n−2 ) = ∞, and so condition (3) fails, as required.  A not completely transparent, aesthetically pleasing example of a metric space for which the real-valued coarse maps form a ring is (−1, 1) ∪ {2n : n ∈ N} as a metric subspace of the real line. To see this, simply note that each unbounded subset must contain 2n for infinitely many n ≥ 1, and I (2n ) = 2n−1 . The set of positive integers as a metric subspace of the real line is a Cabello-Sánchez space–in fact, a UC-space–that does not satisfy condition (1) of Theorem 5.2. On the other hand, any bounded dense-in-itself metric space that is not Bourbaki bounded in itself satisfies condition (1) but is not a Cabello-Sánchez space (see Example 3.5). Turning to stability under reciprocation, the situation is different in that for a given metric space, the class of continuous real-valued coarse maps defined on it can be

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stable under reciprocation while the larger class of real-valued coarse maps need not be. For the purposes of our next result, for a subset A of R, let us agree to write just diam(A) for its diameter with respect to the Euclidean metric. Theorem 5.3 Let X, d be a metric space. The following conditions are equivalent: (1) the real-valued coarse maps on X are stable under reciprocation; (2) ∀t > 0, there are at most finitely many nontrivial subsets of X with diameter at most t. Proof (2) ⇒ (1). First observe that condition (2) implies that each bounded subset of X is finite, for if diamd (A) ≤ t and A is infinite, then the family of subsets of A consisting of exactly two points would be an infinite family of subsets of X of diameter at most t. Suppose f : X → R is nonvanishing and coarse. We intend to show that for each t > 0 there exists Mt > 0 such that whenever A ⊆ X is nonempty and diamd (A) ≤ t, then diam( 1f (A)) ≤ Mt . Let E 1 , E 2 , . . . , E n enumerate the nontrivial subsets of X whose diameters are at most t. For each j ≤ n, (1/ f )(E j ) is a finite set of real numbers and thus has finite diameter which we denote by α j . It follows that if diamd (A) ≤ t, then   1 (A) ≤ max{α1 , α2 , . . . , αn }, diam f and so by Proposition 5.1 the reciprocal is a coarse map. (1) ⇒ (2). Assume condition (1) is satisfied. We first show that if A ⊆ X is a bounded subset of X , then A must be finite. If not, we can find a sequence an  in A with distinct terms. Define f : X → R by f (an ) = n1 and f (x) = 1 otherwise. Being bounded, f is a coarse map, but 1f satisfies ω 1f (diamd (A)) = ∞, so that the reciprocal is not a coarse map. We now assume condition (2) fails to reach a contradiction. Let {E n : n ∈ N} be a countably infinite family of distinct nontrivial subsets of X each of whose diameters is at most t for some positive t. Clearly E := ∪∞ n=1 E n must be infinite. By the finiteness of bounded subsets of X , the set E can’t be bounded. But if the union were unbounded, then E would also be a subset of X such that sup{I (e) : e ∈ E} ≤ t. By Theorem 5.2, the real-valued coarse maps on X are not stable under pointwise product, so by our key result Theorem 1.2, stability under reciprocation also fails.  Example 5.4 Condition (2) of our last result implies that X is countable because each ball in X must be finite. A countably infinite set equipped with the zero-one metric will not satisfy condition (2), but will be a space for which the real-valued coarse maps are stable under pointwise product because each subset is bounded. It is clear that for an arbitrary compact metric space X, d, the continuous realvalued coarse maps coincide with C(X, R), and so in this setting the continuous coarse maps are stable under reciprocation. At the moment, we do not know how to characterize those metric spaces for which the continuous real-valued coarse maps are stable under reciprocation.

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References 1. Aggarwal, M., Cobza¸s, S.: ¸ On some Lipschitz-type functions. J. Math. Anal. Appl. 517, 126631 (2023) 2. Albiac, F., Kalton, N.: Topics in Banach Spaces. Springer, New York (2006) 3. Atsuji, M.: Uniform continuity of continuous functions of metric spaces. Pac. J. Math. 8, 11–16 (1958) 4. Beer, G.: More about metric spaces on which continuous functions are uniformly continuous. Bull. Austral. Math. Soc. 33, 397–406 (1986) 5. Beer, G.: UC spaces revisited. Am. Math. Monthly 95, 737–739 (1988) 6. Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dordrecht, Holland (1993) 7. Beer, G.: McShane’s theorem revisted. Vietnam J. Math. 48, 237–246 (2020) 8. Beer, G., García-Lirola, L., Garrido, M.I.: Stability of Lipschitz-type functions under pointwise product and reciprocation. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 114 (2020) 9. Beer, G., Garrido, M.I.: Bornologies and locally Lipschitz functions. Bull. Aust. Math. Soc. 90, 257–263 (2014) 10. Beer, G., Garrido, M.I.: Locally Lipschitz functions, cofinal completeness, and UC spaces. J. Math. Anal. Appl. 428, 804–816 (2015) 11. Beer, G., Garrido, M.I.: On the uniform approximation of Cauchy continuous functions. Top. Appl. 208, 1–9 (2016) 12. Beer, G., Garrido, M.I.: Real-valued Lipschitz functions and metric properties of functions. J. Math. Anal. Appl. 486, 123839 (2020) 13. Beer, G., Garrido, M.I., Meroño, A.S.: Uniform continuity and a new bornology for a metric space. Set-Valued Var. Anal. 26, 49–65 (2018) 14. Benyamini, Y., Lindenstrauss, J.: Geometric nonlinear functional analysis. In: American Mathematical Society, vol. 48. Colloquium Publications, Providence, Rhode Island (2000) ˇ 15. Borsik, J.: Mappings preserving Cauchy sequences. Casopis pˇest. Mat. 113, 280–285 (1988) 16. Bouziad, A., Sukhacheva, E.: Preservation of uniform continuity under pointwise product. Top. Appl. 254, 132–144 (2019) 17. Braga, B.M.: Coarse and uniform embeddings. J. Funct. Anal. 272, 1852–1875 (2017) 18. Cabello-Sánchez, J.: U(X ) as a ring for metric spaces X . Filomat 31, 1981–1984 (2017) 19. Cobza¸s, S, ¸ Miculescu, R., Nicolae, A.: Lipschitz Functions. Springer, Cham, Switzerland (2019) 20. Frolik, Z.: Existence of ∞ partitions of unity. Rend. Sem. Mat. Univ. Politech. Torino 42, 9–14 (1984) 21. Garrido, M.I., Jaramillo, J.: Lipschitz-type functions on metric spaces. J. Math. Anal. Appl. 340, 282–290 (2008) 22. Garrido, M.I., Meroño, A.S.: New types of completeness in metric spaces. Ann. Acad. Sci. Fenn. Math. 39, 733–758 (2014) 23. Hausdorff, F.: Erweiterung einer Homöomorphie. Fund. Math. 16, 353–360 (1930) 24. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001) 25. Heinonen, J.: Lectures on Lipschitz Analysis. Report, University of Jyväskylä (2005) 26. Jain, T., Kundu, S.: Atsuji completions: equivalent characterizations. Top. Appl. 154, 28–38 (2007) 27. Katetov, M.: On real-valued functions in topological spaces. Fund. Math. 38, 85–91 (1951) 28. Kundu, S., Jain, T.: Atsuji spaces: equivalent conditions. Topol. Proc. 30, 301–325 (2006) 29. Leung, D., Tang, W.-K.: Functions that are Lipschitz in the small. Rev. Mat. Complut. 30, 25–34 (2017) 30. Luukkainen, J.: Rings of functions in Lipschitz topology. Ann. Acad. Sci. Fenn. Series A. I. Math. 4, 119–135 (1978-79) 31. Marino, G., Lewicki, G., Pietramala, P.: Finite chainability, locally Lipschitzian and uniformly continuous functions. Z. Anal. Anwend. 17, 795–803 (1998)

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32. McShane, E.: Extension of range of functions. Bull. Am. Math. Soc. 40, 837–842 (1934) 33. Miculescu, R.: Approximation of continuous functions by Lipschitz functions. Real Anal. Exhange 26, 449–452 (2000-2001) 34. Monteiro, A.A., Peixoto, M.M.: Le nombre de Lebesgue et la continuité uniforme. Portugaliae Math. 10, 105–113 (1951) 35. Rainwater, J.: Spaces whose finest uniformity is metric. Pacif. J. Math. 9, 567–570 (1959) 36. Rice, M.: A note on uniform paracompactness. Proc. Am. Math. Soc. 62, 359–362 (1977) 37. Snipes, R.: Functions that preserve Cauchy sequences. Nieuw Archief Voor Wiskunde 25, 409–422 (1977) 38. Vaughan, H.: On locally compact metrizable spaces. Bull. Am. Math. Soc. 43, 532–535 (1937) 39. Willard, S.: General Topology. Addison-Wesley, Reading, MA (1970)

Lipschitzian Stability in Linear Semi-infinite Optimization M. J. Cánovas and J. Parra

Dedicated to J.C. Ferrando on his 65th Birthday

Abstract This paper is intended to provide an overview of recent results by the authors, together with different collaborators, on quantitative measures of the Lispchitzian behavior of the feasible and the optimal (argmin) sets in continuous linear semi-infinite optimization (with a finite amount of variables and possibly infinitely many constraints). Specifically, the paper focuses on the computation of the global Hoffman constant for the feasible set mapping as well as the Lipschitz and calmness moduli for both the feasible and the optimal set mappings. We point out the fact that all these measures are computed through point-based formulae; i.e., only involving the problem’s data, not appealing to elements in a neighborhood. Hence, the computation of such measures is conceptually implementable in practice. The difficulties appearing in contrast to the finite setting (with finitely many constraints) is highlighted. On that basis, to the authors knowledge, this work presents the state of the art on the point-based computation of the referred measures in the context of linear semi-infinite optimization, with an incursion into infinite-dimensional spaces of variables.

This research has been partially supported by Grant PGC2018-097960-B-C21 from MICINN, Spain, and ERDF, “A way to make Europe”, European Union, and Grant PROMETEO/2021/063 from Generalitat Valenciana, Spain. M. J. Cánovas · J. Parra (B) Center of Operations Research, Miguel Hernández University of Elche, 03202 Elche (Alicante), Spain e-mail: [email protected] M. J. Cánovas e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_4

87

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Keywords Hoffman constants · Aubin property · Calmness · Linear semi-infinite optimization · Feasible set mapping · Optimal set mapping Mathematics Subject Classification 90C31 · 49J53 · 90C34 · 15A39 · 90C05

1 Introduction This paper is focused on Lipschitz-type properties and their corresponding moduli for the feasible and the optimal set mappings in continuous linear semi-infinite optimization. The difficulties arising in the generalization of classical results established in ordinary (finite) linear optimization are highlighted. Specifically, in a first step the paper is intended to present the state of the art on point-based computation (only involving the given data) of the global Hoffman constant and the Lipschitz and calmness moduli of the set-valued mapping F, defined in (2), associated with a linear inequality system. At this moment we advance that this multifunction has a closed and convex graph, and this fact entails remarkable advantages (in particular, Theorems 1 and 2 apply). Secondly, we deal with multifunction S, defined in (3), associated with the parameterized problem (1). The graph of S is no longer convex and this fact entails additional difficulties. In particular, Theorem 2 does not apply and the problem of computing or estimating the global Hoffman constant in our semi-infinite setting remains as open problem; see [8] for additional comments on this subject and, in particular, for the study of a certain directional convexity for finite linear optimization problems. So, regarding multifunction S, the current work focuses on Lipschitz and calmness moduli. Both multifunctions, F and S, are defined in the context of continuous linear semi-infinite programming (LSIP, in brief) under canonical perturbations; i.e., where perturbations fall on the right-hand side of the constraints and the objective function coefficients. Formally, we consider the parameterized linear optimization problem π (c, b) : minimize c x subject to at x ≤ bt , t ∈ T,

(1)

where x ∈ Rn is the vector of decision variables, T is a compact Hausdorff space, a = (at )t∈T is a fixed continuous function from T to Rn and (c, b) ∈ Rn × C (T, R) , with b = (bt )t∈T , is the parameter to be perturbed, C (T, R) being the space of continuous functions from T to R. Elements in Rn are considered as column-vectors and the prime stands for transposition, so that x  y denotes the usual inner product of x and y. The corresponding feasible and the optimal set (argmin) mappings, F : C (T, R) ⇒ Rn and S : Rn × C (T, R) ⇒ Rn , are respectively defined by

Lipschitzian Stability in Linear Semi-infinite Optimization

  F (b) := x ∈ Rn | at x ≤ bt , t ∈ T ,   S (c, b) := arg min c x | x ∈ F (b) .

89

(2) (3)

When T is finite we are placed in the classical framework of linear programming (LP, in brief), whose theoretical analysis comes from the early 1950s (see, e.g., [22, 26, 44]). At this moment, let us point out the Hoffman lemma (cf. [26]), which ensures the existence of some constant κ ≥ 0 such that   d (x, F (b)) ≤ κ max at x − bt + for all x ∈ Rn and all b ∈ dom F, t∈T

(4)

where dom F denotes the domain of F (the set of all b with F (b) = ∅), [α]+ := max {α, 0} is the positive part of α ∈ R, and d (x, F (b)) is the distance (in Rn ) from x to set F (b) . The infimum of constants κ in (4) is called the global Hoffman constant of F, where the terminology ‘global’ follows from thefact that inequality (4)  involves all points x ∈ Rn and all b ∈ dom F. Since maxt∈T at x − bt + coincides   with the distance (in RT with the supremum norm) d b, F −1 (x) , inequality (4) can be written in the variational form   d (x, F (b)) ≤ κd b, F −1 (x) for all x ∈ Rn and all b ∈ dom F,

(5)

which, indeed, has inspired the definition of Hoffman property for a generic multifunction (introduced in [7] and recalled in Sect. 2). In the late 1960s, the analysis of semicontinuity and Lipschitz-type properties based on approaches of variational analysis like Berge’s theory or Hoffman’s error bounds arose and became a research area of growing interest (see, e.g., [3, 18, 20, 30, 32, 34, 40, 41, 45, 46]). Indeed, Lipschitz-type properties are basic quantitative notions in the core of variational analysis which are widely used in both theoretical and computational studies (see the monographs [19, 28, 36, 43]). In contrast with the global nature of the Hoffman constant, the Lipschitz and calmness moduli constitute local measures of the stability of feasible and optimal solutions with respect to parameter perturbations; see again Sect. 2 for the corresponding definitions. Let us emphasize the practical repercussions of inequalities of the form (4) in their global or local versions, forinstance with respect to the convergence of algorithms, as far as the residual maxt∈T at x − bt + in the right-hand side is usually easier to compute or estimate than the distance in the left-hand side. In [39] we can find several references on the algorithmic repercussions of Hoffman constants and other related error bounds in modern convex optimization algorithms. Concerning LSIP, the reader is addressed to the monograph [24] for a comprehensive development of theory, methods, and applications in the case when the index set T is arbitrary, including the particular case of continuous LSIP. Continuity properties of multifunctions F and S in the continuous LSIP setting were originally tackled in [5, 21], while the counterpart results for an arbitrary T can be found in [24]. Section 6 in book [25] constitutes the immediate antecedent to this work, updated to 2014. Indeed, Theorems 5, 7, 9, and 11 can also be found there.

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The structure of this paper is as follows: Sect. 2 gathers the necessary notation, basic definitions, and tools used throughout the paper. Section 3 provides the existing results, up to the authors’ knowledge, on explicit point-based formulae for the global Hoffman constant and the Lipschitz and calmness moduli of F in our current semi-infinite framework. The results about S are concentrated in Sect. 4. Section 5 tackles some extensions to infinite-dimensional spaces of variables. Finally, some conclusions and perspectives are gathered in Sect. 6.

2 Preliminaries Given S ⊂ Rk , k ∈ N, we denote by conv S and cone S the convex hull and the conical convex hull of S, respectively. It is assumed that cone S always contains the zero-vector 0k , in particular cone(∅) = {0k }. Moreover, S ◦ denotes the (negative) polar of S, given by   S ◦ := u ∈ Rk | u  x ≤ 0 for all x ∈ S (S ◦ = Rk if S = ∅). From the topological side, if S is any subset of a topological space, intS, clS and bdS stand for the (topological) interior, closure, and boundary of S, respectively.

2.1 On Generic Multifunctions The definitions given in this subsection can be stated in general metric spaces, although the results provided in this paper are given for (general or particular) Banach spaces. For this reason, hereafter we confine ourselves to the latter framework. Given a multifunction M : Y ⇒ X between Banach spaces with associated distances being denoted by d, we say that the Hoffman property holds if there exists a constant κ ≥ 0 such that   d(x, M(y)) ≤ κd y, M−1 (x) for all x ∈ X and all y ∈ dom M,

(6)

where d (x, ) := inf {d (x, ω) | ω ∈ } for x ∈ X and  ⊂ X, with inf ∅ := +∞, so that d (x, ∅) = +∞. Since this paper is concerned with nonnegative constants, we use the convention sup ∅ := 0. Here dom M is the domain of M (recall that y ∈ dom M ⇔ M(y) = ∅) and M−1 denotes the inverse mapping of M (i.e. y ∈ M−1 (x) ⇔ x ∈ M(y)). In contrast to the previous global Lipschitz-type property, Aubin property is of local nature as inequality (6) is required to be satisfied only in a neighborhood of a given (nominal) element (y, x) ∈ gphM (the graph of M). This property appears in the literature under different names as pseudo-Lipschitz (see, e.g., [28]; indeed, this

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is the original name introduced by Aubin in [1]) or Lipschitz-like (see, e.g., [36]). Formally, we say that M has the Aubin property at (y, x) ∈ gphM if there exists a constant κ ≥ 0 along with a neighborhood of (y, x) , V × U, in the product space Y × X such that   d(x 1 , M(y 2 )) ≤ κd y 1 , y 2 for all x 1 ∈ M(y 1 ) ∩ U and all y 1 , y 2 ∈ V.

(7)

Aubin property of M at (y, x) is known to be equivalent to the metric regularity of its inverse M−1 at (x, y) , which reads as the existence of κ ≥ 0 and a (possibly smaller) neighborhood U × V of (x, y) such that   d(x, M(y)) ≤ κd y, M−1 (x) for all (x, y) ∈ U × V.

(8)

When we fix y = y, we are dealing with the calmness property. The calmness property was called ‘upper Lipschitzian’ by Robinson, who established in [41] its fulfilment for piecewise polyhedral mappings. Specifically, M is said to be calm at (y, x) ∈ gphM if there exist a constant κ ≥ 0 and a neighborhood of (y, x) , V × U, such that d(x, M(y)) ≤ κd (y, y) for all x ∈ M(y) ∩ U and all y ∈ V.

(9)

It is also known that M is calm at (y, x) ∈ gphM if and only if M−1 is metrically subregular at (x, y) ; i.e., if there exist a constant κ ≥ 0 and a (possibly smaller) neighborhood U of x such that   d(x, M(y)) ≤ κd y, M−1 (x) for all x ∈ U.

(10)

Now we recall the definitions of the moduli corresponding to the properties introduced above: • The infimum of constants κ appearing in (6) is called the global Hoffman constant of M and it is denoted by Hof M. • The infimum of constants κ appearing in (7), for some associated neighborhoods, is known to coincide with the infimum of those κ in (8); it is called the Lipschitz modulus of M at (y, x) and denoted by lipM (y, x) ; see e.g., [28, Sects. 1.4 and 1.5] and references therein. • Analogously, the infimum of constants κ in (9) and (10), again for some associated neighborhoods, also coincide; see again [28, Sects. 1.4 and 1.5]. This infimum is called the calmness modulus of M at (y, x) and denoted by clm M (y, x) . As a consequence of the definitions, the three moduli may be written as follows:

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Hof M =

sup

(y,x)∈(dom M)×X

d(x, M(y))  , d y, M−1 (x)

d(x, M(y))  , −1 (y,x)→(y,x) d y, M (x)

lipM(y, x) = lim sup

clm M (y, x) = lim sup x→x

(11)

d(x, M(y))  , d y, M−1 (x)

under the convention 00 := 0, where lim sup is understood as the supremum (maximum, indeed) of all possible sequential upper limits (i.e., with (y, x) being replaced with elements of sequences {(yr , xr )}r ∈N converging to (y, x) as r → ∞). For a closed convex multifunction between Banach spaces, as a consequence of the classical Robinson-Ursescu Theorem and taking into account the equivalence between the Aubin property of M and the metric regularity of M−1 , we have the following result. Theorem 1 (Robinson-Ursescu) Let M : Y ⇒ X with closed and convex graph. Then M has the Aubin property at (y, x) ∈ gphM if and only if y ∈ int domM. In the following lemma X is assumed to be a reflexive Banach space in order to ensure the existence of best approximations on closed convex sets; see e.g. [47, Theorem 3.8.1]. The statement of the lemma constitutes a key step in the proof of the subsequent theorem which establishes the relationship between the global stability measure Hof M and the local ones clm M (y, x) , with (y, x) ∈ gph M. Lemma 1 ([7, Lemma 2]) Let M : Y ⇒ X with a non-empty convex graph, and assume that X is reflexive. Let y ∈ dom M and suppose that M (y) is closed. Consider any (y, x) ∈ gph M and let x be a best approximation of x in M (y), then d (x, M (y)) ≤ clm M (y, x) . d (y, y) Theorem 2 ([7, see Sect. 1 and Theorem 4]) Let M : Y ⇒ X with a non-empty convex graph and closed images. Assume that X is reflexive. Then one has Hof M =

sup

(y,x)∈gph M

clm M (y, x) .

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2.2 On Linear Inequality Systems   Consider our feasible set mapping F introduced in (2) and a nominal element b, x ∈ gphF. We introduce the notation:   Tb (x) := t ∈ T | at x − bt = 0 ,   which is called the subset of active indices of system at x ≤ bt , t ∈ T at the feasible point x; the following set constitutes a key ingredient in several formulae of the aimed moduli:   (12) Cb (x) := conv at , t ∈ Tb (x) . Observe that Tb (x) is a compact subset of T and, consequently, Cb (x) ⊂ Rn is also compact. At this moment, we comment that Cb (x) will appear in the computation of the Lipschitz modulus of F, whereas its end set, end Cb (x) , will appear in the computation of the calmness modulus. Recall that, given a nonempty convex set C ⊂ Rk , its end set, end C (introduced in [27]), is defined as end C := {u ∈ cl C | μ > 1 such that μu ∈ cl C} . with an arbiWith respect to the topology, the space of variables, Rn , is   equipped trary norm, · , with dual norm given by u∗ = maxx≤1 u  x  , while the parameter space C (T, R) is endowed with the supremum norm b∞ := maxt∈T |bt | ; recall that a ≡ (at )t∈T ∈ (Rn )T is fixed. For comparative purposes, the next theorem gathers two expressions which can be found in the literature on Hof F when confined to finite linear systems, where RT ≡ Rm for some m ∈ N. They come from [39, Formulae (3) and (4)] (see [30, Theorem 2.7] when Rn is endowed with the Euclidean norm); one can find an alternative expression based on a dual approach in [6, Theorem 8]. Theorem 6 provides the extension of this result to our semi-infinite framework. Theorem 3 Consider the feasible set mapping F defined in (2) and assume that T is finite. We have Hof F = =

max

J ⊂T / 0n ∈conv{a t , t∈J }

max

d∗ (0n , conv {at , t ∈ J })−1

J ⊂T, rank A J =rank A {at , t∈J } lin. indep.

d∗ (0n , conv {at , t ∈ J })−1

(13) (14)

where A J and A stand for the matrices whose rows are at , with t ∈ J and t ∈ T, respectively, and d∗ stands for the distance associated with the dual norm ·∗ . Many other authors have contributed to the study of Hoffman constants and their relationship with other concepts (as Lipschitz constants). Additional references can be obtained from [2, 48], among others. At this moment we also cite [4, 34, 41].

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Regarding  the calmness modulus, the following theorem provides a formula for  clm F b, x through points outside F(b). It appeals to the supremum function f b : Rn → R, given by   f b (x) := sup at x − bt , for x ∈ Rn , t∈T

which is known to be convex on Rn . For each x ∈ Rn , we consider the subset of indices   Jb (x) = t ∈ T | at x − bt = f b (x) . The well-known Valadier’s formula works by virtue of the Ioffe-Tikhomirov theorem (see e.g. [47, Theorem 2.4.18]), yielding   ∂ f b (x) = conv at , t ∈ Jb (x) , where ∂ f b (x) stands for the usual subdifferential of convex analysis (see e.g. [42]).   Theorem 4 ([31, Theorem 1]) Given b, x ∈ gph F, one has clmF(b, x) = =

 −1 lim sup d∗ 0n , ∂ f b (x)

x→x, f b (x)>0

  −1 lim sup d∗ 0n , conv at , t ∈ Jb (x) .

x→x, f b (x)>0

Remark 1 In the terminology of [31], (clmF(b, x))−1 is the error bound modulus (also known as conditioning rate in [38]) of f b at x. In relation to Theorem 4, our objective is to present an alternative formula for the involved calmness modulus which only appeals to the nominal elements. To this respect, let us comment that the problem is solved in [17, Theorem 4] for finite systems and its extension to semi-infinite systems under an appropriate regularity condition was established by Li, Meng, and Yang in [35, Corollary 2.1, Remark 2.3 and Corollary 3.2]. This result is recalled in Theorem 8. Finally, regarding the Aubin property of F, the following theorem recalls different characterizations which are well-known in the literature (see, [24, Theorem 6.1] for the extension to the case when T is arbitrary). In particular, equivalence (i) ⇔ (ii) in this theorem is nothing else but the Robinson-Ursescu Theorem (see Sect. 2.1), taking into account that gphF is closed and convex. Condition (iii) in Theorem 5 appeals to the Slater condition which is satisfied at b when there exists a strict solution -called Slater point- of the associated linear system (i.e., when there exists x < bt , for all t ∈ T ). Observe that in our current continuous setting (T x such that at being compact Hausdorff and t → (at , bt ) being continuous) the Slater condition is equivalent to the so-called strong Slater condition (for  arbitrary  T and t → (at , bt )) x − bt < 0. which reads as the existence of x such that supt∈T at

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Theorem 5 Let (b, x) ∈ gphF. The following conditions are equivalent: (i) F has the Aubin property at (b, x); (ii) b ∈ intdomF ; (iii) b satisfies the Slater condition; / Cb (x) . (iv) 0n ∈

3 Lipschitzian Behavior of the Feasible Set This section brings together point-based formulae for three stability measures (global Hoffman constant, Lipschitz modulus, and calmness modulus) in the context of linear inequality systems. Roughly speaking, the global Hoffman constant provides information on the rate of variation of feasible solutions with respect to parameter perturbations. Observe that, by definition, this measure deals always with consistent systems since inequality (6) in the particular case M := F is only required for parameters b ∈ domF. In the Lipschitz modulus is focussed on the local   contrast, (around a nominal pair b, x ∈ gphF) rate of variation of solutions with respect to parameters without restricting them to the domain of F. Hence, lipF(b, x) < ∞ entails b ∈ int domF (indeed, both conditions are equivalent as it is stated in Theorem 5). So, again roughly speaking, lipF(b, x) controls both the rate of local enlargement and the rate of local shrinkage. Finally, regarding the definition of clmF(b, x), observe that it only controls the enlargement   of  the feasible set around the nominal x as far as we are just estimating d x, F b for nearby feasible solutions x of parameters closed to b, which remains fixed. To start with, the following very recent result extends Theorem 3 above to our current semi-infinite context. In it, we additionally require the fact that T is a metric compact space (in order to ensure that any closed subset of T is exactly the zero level set of a nonnegative continuous function on T ). Theorem 6 ([7, Theorem 5]) Assume that T is a metric compact space. Consider F : C (T, R) ⇒ Rn defined in (2). We have Hof F =

sup J ⊂T compact 0n ∈conv{a / t , t∈J }

d∗ (0n , conv {at , t ∈ J })−1 .

With respect to the Lipschitz modulus of F, the following theorem provides a formula in terms of the (dual) distance from 0n to Cb (x) . Observe that it covers the case when F has not the Aubin property at (b, x), since, according to Theorem 5, this is equivalent to the fact that 0n ∈ Cb (x) , in which case (15) yields lipF(b, x) = 01 = +∞. On the other hand, if x is a Slater point for b, then Cb (x) = ∅ and lipF(b, x) = 1 = 0. ∞

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Theorem 7 ([9, Corollary 3.2]) Let (b, x) ∈ gphF. Then, one has  −1 . lipF(b, x) = d∗ 0n , Cb (x)

(15)

The rest of this section is devoted to the calmness property. This property is always fulfilled for finite systems since, when T is finite, F is a polyhedral mapping and the statement comes from the classical work of Robinson [41], as commented in Sect. 2.1. In contrast, the analysis of the calmness property in the semi-infinite framework becomes much more involved. Theorem 2.2 in Zheng and Ng [49] characterizes this property in terms of the so-called strong basic constraint qualification at x, which reads as the existence of τ > 0 and a neighborhood U of x such that N (F(b), x) ∩ B∗ ⊂ [0, τ ]∂ f b (x), for all x ∈ U ∩ bdF(b),

(16)

where N (F(b), x) represents the usual normal cone (of convex analysis in Rn ) to F(b) at x and B∗ stands for the closed unit ball with respect to the dual norm. Theorem 3 in [17] provides an alternative characterization in terms of the Abadie constraint qualification together with the uniform dual boundedness condition introduced in that paper (see [17, Definition 2]). The computation of the calmness modulus of F at (b, x) is completely determined in [17, Theorem 4] for finite systems and its extension to the semi-infinite setting entails remarkable difficulties. Indeed, to the authors knowledge, it remains as an open problem in our continuous linear semi-infinite framework unless some additional assumption is required. Specifically, Theorem 8, due to Li et al. [35] provides the extension of [17, Theorem 4] to semi-infinite systems under the following regularity condition at x: “There exists a neighborhood W of x such that   F(b) ∩ W = x + Ab (x)◦ ∩ W ”,

(17)

where Ab (x) is the corresponding active cone at x; i.e.,   Ab (x) := cone at , t ∈ Tb (x)

(18)

(recall that Ab (x) = {0n } if Tb (x) = ∅). Observe that this condition is held at all points of polyhedral sets and, for instance, at the vertex of the ice-cream cone, for appropriate linear representations. Indeed, the fulfilment of condition (17) at all points of F(b) is equivalent to the fact that the associated linear inequality system is locally polyhedral (see [35, Corollary 3.3] and also [24, Sect. 5.2]). In order to derive the aimed point-based formula for clmF(b, x), the following family, Db (x) (introduced in [17, Sect. 4]), of subsets D ⊂ Tb (x) plays a crucial role. By definition, D ∈ Db (x) if the system

at d = 1, t ∈ D, at d < 1, t ∈ Tb (x) \ D

 (19)

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97

is consistent in the variable d ∈ Rn . Observe that if D ∈ Db (x) and d is such a solution, then {at , t ∈ D} is contained in the hyperplane {z ∈ Rn | d  z = 1}, which leaves {0n } ∪ {at , t ∈ Tb (x) \ D} on one of its two associated open half-spaces. Theorem 8 ([35, Corollary 2.1, Remark 2.3, and Corollary 3.2]) Let x ∈ F(b) and assume that the regularity condition (17) is held at x. Then    −1 clm F b, x = d∗ 0n , endCb (x) =

(20)

sup d∗ (0n , conv {at , t ∈ D})

D∈Db (x)

−1

.

Observe that  if x is a1 Slater point, endCb (x) = ∅ (since Cb (x) = ∅), and (20) = 0. yields clm F b, x = +∞ Remark 2 Let us point out that the second equality of Theorem 8 holds without requiring the regularity condition (17). Indeed, from [35, Corollary 2.1 and Remark 2.3] we can deduce D∈Db (x)

conv {at , t ∈ D} ⊂ endCb (x) ⊂ cl

D∈Db (x)

 conv {at , t ∈ D} .

However, condition (17) is not superfluous regarding the first equality, (20), as Example 1 shows (it comes from modifying [17, Example 1] and was revisited in [35, Example 3.3]). Example 1 Let us consider the system, in R2 endowed with the Euclidean norm, given by ⎫ ⎧ ⎨ t (cos t) x1 + t (sin t) x2 ≤ t, t ∈ [0, π] , ⎬ x1 ≤ 1, t = 4, ; ⎭ ⎩ −x1 − x2 ≤ 1, t = 5 i.e., T := [0, π] ∪ {4, 5}, at := t (cos t, sin t) , for t ∈ [0, π] , a4 := (1, 0) and a5 := (−1, −1) ; b ∈ C ([0, π] ∪ {4, 5}, R) is given by bt = t, t ∈ [0, π] , b4 = 1, and b5 = 1. Consider the feasible points x 1 = (1, 0) and x 2 = (1, −2) . As proved in [17, Example 1], we have that   clm F x 1 , b = +∞.  cos 1   Alternatively, we can apply Theorem 4 with the sequence x r = 1 + r1 sin 1r . It r

is clear at x1 . Indeed (0, 1) ∈  1 ◦that the regularity  condition (17) is 1not satisfied  ◦  = cone (1, 0) = R− × R, but x + ε (0, 1) ∈ / F b for any ε > 0. Ab x Moreover,     Cb x 1 = conv (0, 0) , (1, 0)       −1   . and, so, endCb x 1 = (1, 0) . Hence clm F x 1 , b = d 02 , endCb x 1

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Fig. 1 Illustration of Example 1

2 With respect (17) is satisfied  2to ◦ point  2  at  x , 2one easily sees that condition  x = A = u ∈ R | −u − u ≤ 0, u ≤ 0 . In this case, C x , where 2 1 b b x  1  conv (0, 0) , (1, 0) , (−1, −1) . Hence, from Theorem 8 we have (Fig. 1) 2

 −1    clm F x 2 , b = d 02 , endCb x 2 −1 √   = d 02 , conv (1, 0) , (−1, −1) = 5.

4 Lipschitzian Behavior of the Argmin Mapping This section is focused on the Lipschitz and calmness moduli of S : Rn × C (T, R) ⇒ Rn introduced in (3). Recall that we are considering an arbitrary norm, · , in the space of variables Rn (whose dual norm is denoted by ·∗ ). The parameter space, Rn × C (T, R) , is endowed with the maximum norm (c, b) := max {c∗ b∞ } . As announced in Sect. 1, the computation of the global Hoffman constant of S still remains as open problem. Let us start by analyzing the Aubin property of this multifunction. In this setting,evenin more general contexts (see [28, Corollary 4.7]), the Aubin  property of S at c, b , x ∈ gphS turns out to be equivalent to the so-called strong Lipschitz   stability, which reads as follows: There exist neighborhoods U of x and V of c, b and a constant κ ≥ 0 such that S(c, b) ∩ U is a singleton, {x(c, b)}, for all (c, b) ∈ V and

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99

    1 1 x(c , b ) − x(c2 , b2 ) ≤ κ (c1 , b1 ) − (c2 , b2 ) whenever (c1 , b1 ), (c2 , b2 ) ∈ V. Note that, because of the convexity of S(c, b), indeed we have S(c, b)= {x(c,  b)} for all (c, b) ∈ V . In other words, the strong Lipschitz stability of S at c, b , x is   equivalent to local single-valuedness and Lipschitz continuity of S at c, b . With the aim of establishing a point-based characterization of the Aubin property of S, which, going further, helps to provide a point-based formula for its Lipschitz modulus, the following   family of subsets of indices constitutes a key ingredient: Associated with ( c, b , x) ∈ gphS, we consider the family of minimal KKT subsets given by    − c ∈ cone({at , t ∈ D}),  . Mc,b (x) = D ⊂ Tb (x)  D is minimal for the inclusion order

The ‘KKT’ terminology comes from the well-known Karush-Kuhn-Tucker optimality condition (KKT condition, for short), which is held at x ∈ F(b) when −c ∈ Ab (x) (recall (18)). It is well known that this condition is sufficient to ensure optimality of x and that it is also necessary under the Slater condition at b. Moreover, taking into account (iii) ⇔ (iv) in Theorem 5, when b satisfies the Slater condition, Ab (x) is closed since it is the cone generated by a compact convex set, Cb (x) , which does not contain the origin (see [42, Corollary 9.6.1]). Remark 3 The family Mc,b (x) was introduced in [13, p. 945] as a crucial ingredient in the computation of the calmness modulus of S, although it was implicitly used in previous works. Now, under the current perspective, we are revisiting some results on the Aubin property and the Lipschitz modulus published some years ago (before [13]), and we are rewriting them in terms of Mc,b (x) , which entails a double advantage: a clearer formulation of the results and an easier way to compare Lipschitz versus calmness moduli. In the following theorem, |D| denotes the cardinality of set D.    Theorem 9 ([14, Theorem 16]) Given any c, b , x ∈ gphS, the following conditions are equivalent:    (i) S has the Aubin property at c, b , x ;    (ii) S is strongly Lipschitz stable at c, b , x ;   (iii) S is locally single-valued and continuous in some neighborhood of c, b ;   (iv) S is single-valued in some neighborhood of c, b ; (v) b satisfies the Slater condition and there is no D ⊂ Tb (x) with |D| < n such that −c ∈ cone({at , t ∈ D}); (vi) b satisfies the Slater condition and all D ∈ Mc,b (x) satisfies |D| = n. Remark 4 Theorem 9(v) is also equivalent to a certain condition introduced by Nürnberger [37] (see [14, Theorem 16] for details) which, translated into our framework, reads as follows: b satisfies the Slater condition and for each D ⊂ Tb (x) with

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|D| = n such that −c ∈ cone({at , t ∈ D}) , we have that all the possible subsets with n elements of {at , t ∈ D} ∪ {−c} are linearly independent. According to the previous remark, condition  in  Theorem 9 is referred to in  (v) several papers as Nürnberger’s condition at c, b , x , and hereafter we shall adopt    also this terminology, abbreviated by NC at c, b , x . Remark 5 One can easily check by a standard argument of linear algebra (in the line of Carathéodory’s theorem) that {at , t ∈ D} is linearly independent whenever D ∈ Mc,b (x) (see, e.g., [23, Remark 3]). Hence, under NC, any set of vectors {at , t ∈ D}, with D ∈ Mc,b (x) , forms a basis of Rn ; in other words any matrix A D , whose rows are at , with t ∈ D, is nonsingular when D ∈ Mc,b (x). The norms of A−1 , with  D   D ∈ Mc,b (x) , are the ingredients in the computation/estimation of lipS c, b , x as it is stated in Theorem 10.    The computation of lipS c, b , x is completely determined when T is finite (see Theorem 10); indeed, it was published in [10, Corollary 2], solving  an open problem pointed out in [33, p. 38]. However, the computation of lipS c, b , x by a point-based formula still remains as open problem in the general continuous LSIP setting. This problem was solved in the same paper [10] under a quite technical hypothesis, referred to as condition (H), which we recall below for completeness (see [10, p. 528]). We advance that this condition is always held for dimensions n ≤ 3, while it was shown in [10, Example 3] that it may fail for n = 4. Condition (H): “If T1 , T2 , T3 ⊂ Tb (x) are such that −c ∈ [cone({at , t ∈ T1 ∪ T2 }) ∩ cone({at , t ∈ T1 ∪ T3 })]\cone({at , t ∈ T1 }) 1 ⊂ T1 such that then, there exists a subset T 1 ∪ T2 }) ∩ cone({at , t ∈ T 1 ∪ T3 }).” 1 | ≤ n − 1 and − c ∈ cone({at , t ∈ T |T Hereafter, we will also appeal to the argmin mapping in the context of right-hand side perturbations Sc : C (T, R) ⇒ Rn defined as Sc (b) := S((c, b) , b ∈ C (T, R) , where c remains fixed. Observe that the first equality in (21) says that, roughly speaking, perturbations of c are negligible when computing the Lipschitz modulus of S. Theorem 4 and 7]) Let  2, Proposition    10 ([10, Corollary 2, Theorems 1 and c, b , x ∈ gphS and assume that NC holds at c, b , x ). Then, lipS

   c, b , x = lipSc (b, x) ≥ =

sup

D∈Mc,b (x)

sup

D∈Mc,b (x)

 −1  A  D

  −1 d∗ 0n , bd conv ±at , t ∈ D .

(21)

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If T is finite or condition (H) is satisfied (in particular, when n ≤ 3) the inequality ‘≥’ in (21) holds as an equality. Remark 6 The equality in the second row of (21) comes from the following standard argument: For D ∈ Mc,b (x) we can identify matrix A D with the ‘endomorphism’ Rn  x → A D x ∈ R D , where Rn is equipped with an arbitrary norm · and R D is endowed with the supremum norm ·∞ . For our choice of norms we have   −1       A  := max  A−1 y  = d∗ 0n , bd conv ±a , t ∈ D −1 . t D D y∞ ≤1

(22)

From now on we on the calmness of S. Observe that, when T is finite,   arefocused S is calm at any c, b , x ∈ gphS, since gphS is finite union of polyhedral sets, hence the classical result by Robinson [41] applies. The semi-infinite case entails remarkable differences. To start with, generically, the KKT condition characterizes optimality in LSIP under some constraint qualification, as for instance the Slater condition. Indeed, this fact is behind the requirement of the Slater condition in the following theorem, which characterizes the calmness of S in terms of the calmness of Sc¯ and also of the calmness of the level set mapping L : R×C (T, R) ⇒ R given by   (23) L (α, b) = x ∈ R p | c¯ x ≤ α; at x ≤ bt , t ∈ T . Here we point out the fact that L is nothing else but a feasible set mapping associated to a linear system parameterized with respect to the right-hand side; in other words, it is a multifunction of the type defined in (2) and already analyzed in Sect. 3. ¯ x) ∈ gphS and Theorem 11 ([11, Theorem 3.1 and Remark 3.1]) Let ((c, ¯ b), assume that b satisfies the Slater condition. Then the following conditions are equivalent: ¯ x); (i) S is calm at ((c, ¯ b), ¯ x); (ii) Sc¯ is calm at (b, ¯ x). (iii) L is calm at ((c¯ x, b), Moreover, one has ([15, Theorem 11 and p. 39]) ¯ x) ≤ max clmS((c, ¯ b),



 c ¯ ∗ ¯ x), , 1 clmL((c¯ x, b), αb¯ (x)

where α b¯ (x) := supz=1 inf t∈T b¯ (x) at z > 0. Let us comment that, in the particular case when c = 0, the three equivalent conditions of Theorem 11 are, indeed, always fulfilled, as it is shown in [11, Remark 3.1]. Remark 7 Paper [29] shows that implication (iii) ⇒ (i) above, i.e., calmness of the level set mapping implies calmness of the argmin mapping, may be extended to a wide class of parametric nonlinear programs. However, implication (i) ⇒ (iii) may

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fail if we just replace linear functions in either the objective or the constraints with convex quadratic ones; see [29, Examples 3.1 and 3.2]. The rest of this section is devoted to establish a point-based lower estimate on ¯ x). We advance that this lower estimate equals the modulus when T is clmS((c, ¯ b), finite. For any D ∈ Mc,b (x) we consider the mapping L D : C (T, R) × R D ⇒ Rn given by   L D (b, d) := x ∈ Rn | at x ≤ bt , t ∈ T ; − ai x ≤ di , i ∈ D .

(24)

Observe that     ¯ −b¯ D = x ∈ Rn | at x ≤ bt , t ∈ T ; − ai x ≤ −bi , i ∈ D L D b,   ⊆ S c, ¯ b¯ for all D ∈ Mc,b (x) .

(25)

  ¯ −b¯ D is formed by feasible The previous inclusion ‘⊆’ comes from the fact L D b, points where KKT conditions are satisfied (called KKT points) via the subset of ¯ −b¯ D are optimal. The active indices D; hence, it is clear that all points in L D b, reciprocal is also true as consequence of the minimality of D ∈ Mc,b (x) . Formally, this inclusion holds, in fact, as an equality, as it is said below. This equality is used ¯ x) is to derive the subsequent theorem, where the lower estimate on clmS((c, ¯ b), established. Let us comment that this result can be traced out from [13, Theorem 4.2], although a preliminary version can be found in [15, Theorem 6], where two additional assumptions were required: the uniqueness of optimal solution together with the Slater condition.    Proposition 1 ([13, Proposition 4.1]) Let c, ¯ b¯ , x ∈ gphS. Then     L D b, −b D = S c, ¯ b¯ for all D ∈ Mc,b (x) . Theorem 12 ([13, Theorems 4.1 and 4.2]) Let clmS

     ¯ x ≥ c, ¯ b¯ , x ≥ clmSc¯ b,

   c, ¯ b¯ , x ∈ gphS. Then

sup

D∈Mc,b (x)

clmL D

   ¯ −b¯ D , x , b,

and equalities hold when T is finite. As it is underlined in [13, Sect. 5], to the authors knowledge, the question of whether inequalities in the previous theorem hold as equalities in the continuous linear semi-infinite setting remains as an open problem. Finally,let us comment   that under appropriate assumptions the calmness modulus ¯ −b¯ D , x can be computed through the formula of Theorem 8 as far of L D at b, as L D can be seeing as a new feasible set mapping of the form (2) associated to the index set formed by  cardinality is |D| .  the union of T and a finite set whose ¯ −b¯ D also is (it only adds a In particular, when F b is locally polyhedral, L D b, finite amount of linear inequalities), and we have

Lipschitzian Stability in Linear Semi-infinite Optimization

clmL D

103

      ¯ −b¯ D , x = d∗ 0n , end conv at , t ∈ Tb (x) ; −at , t ∈ D −1 . b,

The following corollary and the subsequent example are intended to show the relationship between the Lipschitz and calmness moduli in the case when T is finite, where exact formulae for both are established in Theorems 10 and 12.    Corollary 1 Let c, ¯ b¯ , x ∈ gphS and assume that T is finite. Then,    ¯ b¯ , x , one has (i) If NC is satisfied at c, lipS

   c, b , x = =

max

  −1 d∗ 0n , bd conv ±at , t ∈ D

max

 −1  d∗ 0n , end conv ±at , t ∈ D .

D∈Mc,b (x) D∈Mc,b (x)

(ii) We have clmS



  c, b , x =

max

D∈Mc,b (x)

  −1 d∗ 0n , end conv at , t ∈ Tb (x) ; −at , t ∈ D .

Remark 8 The last equality of (i) comes from the following argument: It is easy to prove that under NC one has, by applying [24, Theorem A.7], −c ∈ int cone {at , t ∈ D} for all D ∈ Mc,b (x)   and, then, 0n ∈ int conv ±at , t ∈ D . Accordingly, the distance from 0n to the boundary of the latter set equals the distance to its end set. Example 2 ([12, Example 3.1]) Consider the linear optimization problem in R2 endowed with the Euclidean norm, minimize x1 + 13 x2 subject to −x1 ≤ 0, −x1 − 21 x2 ≤ 0, −x1 − x2 ≤ 0, −x1 + x2 ≤ 0,

(t (t (t (t

= 1) = 2) = 3) = 4)

whose unique optimal solution is x¯ = 02 , and where Mc,b (x) = {{1, 2} , {1, 3} , {2, 4} , {3, 4}} .    Since all subsets of Mc,b (x) have cardinality 2, NC holds at c, b , x . One can easily check that the maximum in Corollary 1(i) is attained at D = {1, 2} , so, lipS

   c, b , x =

√ 1   = 17.      −1 , ± −1/2 d∗ 02 , bd conv ± −1 0

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Fig. 2 Illustration of Example 2

√ See Fig. 2, where the previous distance (1/ 17) is the one from the origin to the segments with discontinuous trace. Regarding the calmness modulus, the maximum of Corollary 1(ii) is attained at both D = {1, 2} and D = {1, 3} , and therefore if for instance we choose D = {1, 2}, we can write clmS

   c, b , x =

√ 1   = 5.          −1 , ± −1/2 , −1 , −1 d∗ 02 , bd conv ± −1 0 −1 1

√ See again Fig. 2, where the previous distance (1/ 5) is the one from the origin to the boundary (with continuous trace) of the convex hull of the six points illustrated there.

5 An Incursion into Infinite-Dimensional Spaces of Variables The results of this section are taken from [16], where a coderivative approach is followed to analyze the Aubin property, and its associated Lipschitz modulus, of the feasible set mapping F : ∞ (T ) ⇒ X , given by     F ( p) := x ∈ X | at∗ , x ≤ bt + pt , t ∈ T , p = ( pt )t∈T ,

(26)

where: • Set T indexing the constraints is an arbitrary set (possibly infinite and without any topological structure).

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105

• The parameter space, ∞ (T ) , is formed by all real-valued bounded functions p : T → R and it is endowed with the supremum norm,  p∞ := sup | pt | whenever p = ( pt )t∈T ∈ ∞ (T ) . t∈T

Observe that, for convenience, we are defining the feasible set mapping with perturbations around p = 0T , the zero function on T. • The space of variables, X , is an arbitrary Banach space, whose topological dual is denoted by X ∗ . For simplicity, we use the same notation for the given norm in X and the corresponding dual norm in X ∗ defined as  ∗    x  := sup x ∗ , x | x ≤ 1 , x ∗ ∈ X ∗ , where ·, · denotes the canonical pairing between X and X ∗ . • Elements at∗ ∈ X ∗ , t ∈ T, are fixed and arbitrary, and function b = (bt )t∈T ∈ RT is also fixed and arbitrary (not necessarily bounded). Roughly speaking, we consider bounded perturbations of the possibly unbounded b. From now on, for simplicity in the notation and since there is no ambiguity, all zero vectors (in all spaces) are denoted just by 0. In particular, our nominal parameter will be p = 0 (in ∞ (T )). The coderivative of F at (0, x) ∈ gphF is the positively homogeneous multifunction D ∗ F (0, x) : X ∗ ⇒ ∞ (T )∗ defined by   D ∗ F (0, x) (x ∗ ) := p ∗ ∈ ∞ (T )∗ | ( p ∗ , −x ∗ ) ∈ N ((0, x) ; gphF) , x ∗ ∈ X ∗ , (27) where N (·; ), in general, denotes the set of generalized normals to set  at a given point known as the basic, or limiting, or Mordukhovich normal cone; see, e.g., [36, 43] and the references therein. Since gph F is convex, in this particular case, the basic normal cone appearing in (27) reduces to  

 ( p ∗ , x ∗ ), ( p, x − x) ≤ 0, . N ((0, x); ¯ gph F) = ( p ∗ , x ∗ ) ∈ ∞ (T )∗ × X ∗  ( p, x) ∈ gph F When both ∞ (T ) and X are finite-dimensional, it is known that (see e.g. [43, Theorem 9.40]) F has the Aubin property at (0, x) ∈ gph F if and only if D ∗ F (0, x) (0) = {0},

(28)

and the Lipschitz modulus is computed by        lipF (0, x) =  D ∗ F (0, x) := sup  p ∗  | p ∗ ∈ D ∗ F (0, x) (x ∗ ), x ∗  ≤ 1 . (29) Indeed, the characterization of the Aubin property and the expression of the Lipschitz modulus via coderivatives hold for any closed-graph mapping in finite dimensions

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(see again [43, Theorem 9.40]). In contrast, the infinite dimensional framework is significantly more involved. In [36, Theorem 4.10] we can find a specific version of the characterization of Aubin property in the line of (28) under the assumption that both spaces are Asplund spaces. However, when T is infinite, ∞ (T ) is never Asplund. Moreover, an expression of the type (29) is obtained if, in addition, the space where the multifunction is defined (in our case, ∞ (T )) is finite dimensional. Consequently, we cannot apply the general theory to our multifunction. Nevertheless, we advance that through ad hoc techniques we succeed to state the infinite dimensional counterparts of (28) and (29) for our specific multifunction. The following theorem, which constitutes the infinite dimensional version of Theorem 5, appeals to the strong-Slater condition, which is held  at p = 0 if there exists x ∈ X (called a strong Slater point) such that supt∈T at∗ , x − bt < 0 (see Sect. 2.2 in the particular case X = Rn ). It also appeals to set       C (x) := x ∗ ∈ X ∗ | x ∗ , x ∗ , x ∈ cl∗ conv (at∗ , bt ), t ∈ T , where ‘cl∗ ’ denotes the closure in the weak∗ topology and ‘conv’ represents the convex hull (of elements in X ∗ × R). Roughly speaking, C (x) constitutes the counterpart of Cb (x) (the convex hull of the at ’s corresponding to active indices) in the current setting. For comparative purposes, recall that condition (iv) in Theorem 5 / Cb (x) , which is the translation of Theorem 13 (iv) to the continuous LSIP says 0n ∈ setting of the previous sections. Indeed, one can easily check that C (x) reduces to Cb (x) when confined to the continuous LSIP setting. Nevertheless, as a remarkable difference, let us comment that C (x) does not appeal to active indices. In fact, outside the continuous setting, even maintaining X = Rn , the set of active indices may not play a decisive role. For example, consider the system in R, {r x ≤ 1, r = 1, 2, . . .}, whose feasible set is ] − ∞, 0]; in this case x = 0 is in the boundary of the feasible set, but there are no active indices. Indeed, 0 is a strong Slater point. Finally, observe that Theorem 13(v) establishes the announced coderivative characterization of the Aubin property of F, which extends (28) to our infinite dimensional setting. Theorem 13 ([16, Lemma 2.3 and Theorem 4.1]) Let us consider F : ∞ (T ) ⇒ X defined in (26) and let (0, x) ∈ gph F. The following conditions are equivalent: (i) F has the Aubin property at (0, x); (ii) 0 ∈ int domF ; (iii) 0 satisfies the strong Slater condition; / C (x) ; (iv) 0 ∈ ¯ = {0}. (v) D ∗ F(0, x)(0) The last result ofthis section provides the aimed formula for lipF (0, x), under the assumption that at∗ , t ∈ T is a bounded subset of X ∗ . Again, the role played by Cb (x) in Theorem 7 is now played by C (x). From [16, Lemma 3.4] we have that C (x) = ∅ if x is a strong Slater point at 0. Moreover, if x is not a strong Slater point and w ∗ -compact (in the weak∗ topology) at X ∗ provided at 0, then C (x) is nonempty  ∗ ∗ that set at , t ∈ T ⊂ X is bounded. For comparative purposes we denote as d∗ the distance in X ∗ (with respect to the dual norm).

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107

Theorem 14 ([16, Theorem 4.6 and Corollary 4.7]) Let (0, x) ∈ gph F. Assume the strong  Slater condition is satisfied at p = 0 and that the coefficient set that at∗ , t ∈ T ⊂ X ∗ is bounded. We have: (i) If x is a strong Slater point at 0, then lipF (0, x) = 0. (ii) If x is not a strong Slater point at 0, then   lipF (0, x) =  D ∗ F (0, x) = d∗ (0, C (x))−1 , and the distance d∗ (0, C (x)) is attained.

6 Conclusions and Perspectives The following diagrams are intended to summarize the main results on point-based formulae for the global Hoffman constant of F and the Lipschitz and calmness moduli of F and S. Recall that F and S denote the feasible set mapping and the argmin mapping, defined respectively in (2) and (3), under simultaneous perturbations of the right-hand side of the constraints and the objective function. We also appeal to Sc , the argmin mapping when fixing the objective function x → c x. At this moment we recall the main ingredients in relation to the Lipschitz and calmness moduli of F:   • If Tb (x) denotes the subset of active indices for b at x ∈ F b ,   Cb (x) := conv at , t ∈ Tb (x) . • Family Db (x) of subsets D ⊂ Tb (x) is defined as: D ∈ Db (x) if there exists d ∈ Rn such that at d = 1 for all t ∈ D and at d < 1 whenever t ∈ Tb (x) \ D. diagram 1: results on F Hof F

=

(∗)

sup J ⊂T compact 0n ∈conv{a / t , t∈J }

d∗ (0n , conv {at , t ∈ J })−1

 −1 lipF(b, x) = d∗ 0n , Cb (x)    −1 clm F b, x = d∗ 0n , endCb (x) (∗∗)

=

sup d∗ (0n , conv {at , t ∈ D})

D∈Db (x)

(*) Assuming that T is a metric space. (**) Under the regularity condition (17), held when T is finite. This condition is not superfluous when T is infinite, as Example 1 shows.

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Now we recall key tools regarding S: • Family Mc,b (x) of minimal KKT subsets is formed by all subsets D ⊂ Tb (x) such that − c ∈ cone({at , t ∈ D}) and D is minimal for the inclusion order. • Associated to each D ∈ Mc,b (x) , mapping L D is given by   L D (b, d) := x ∈ Rn | at x ≤ bt , t ∈ T ; − ai x ≤ di , i ∈ D . Observe that L D is a multifunction of the same form as F and all the results in Diagram 1 can be applied.    In the following diagram, formulae related to lipS c, b , x are established    under the Aubin property of S at c, b , x , characterized in Theorem 9. diagram 2: results on S lipS

   c, b , x

= lipSc (b, x)   −1 ≥ sup d∗ 0n , bd conv ±at , t ∈ D (♦) D∈Mc,b (x)   −1 = sup d∗ 0n , end conv ±at , t ∈ D D∈Mc,b (x)

clmS

     ¯ x c, ¯ b¯ , x ≥ clmSc¯ b, (♦♦)

≥

(♦♦)

=

(♦♦♦)

   ¯ −b¯ D , x clmL D b, D∈Mc,b (x)  −1

at , t ∈ Tb (x) , max d∗ 0n , end conv −at , t ∈ D D∈Mc,b (x) sup

(♦) Equality holds when T is finite or condition (H) is satisfied. This condition (H), recalled in Sect. 4, is always held when n ≤ 3, while it may fail for higher dimensions, as it is shown in [10, Example 3]. (♦♦) Equality holds when T is finite. (♦♦♦) Assuming that T is finite. We finish the work with some conclusions and perspectives: • When confined toLP setting Hof F, lipF(b, x),   (T finite),  all  measures  ¯ b¯ , x are completely determined by clm F b, x , lipS c, b , x and clmS c, (conceptually) implementable procedures as far as all these measures can be computed via a finite amount of distances from 0n to finite unions of convex hulls of finitely many points. • In the continuous LSIP setting, Hof F and lipF(b, x) are still completely determined through point-based formulae (in the case of Hof F we are assuming that T is a compact metric space). Going further, let us point out that Theorem 14

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provides the natural generalization of the computation of lipF(b, x) to the infinite dimensional setting;  extension is done via a coderivative approach.  this • In contrast, clm F b, x is established under a quite restrictive regularity condition and the problem of weakening this condition could constitute a future line of research, which would be decisive for computing the calmness modulus of the level set  mapping   introduced in (23), and this measure leads to an upper bound for clmS c, ¯ b¯ , x as it is established in Theorem 11.      • lipS( c, b , x) and clmS c, ¯ b¯ , x are lower bounded by the constants appearing in Diagram 2. The question of whether equalities are held under weaker assumptions than the ones mentioned there remains as an open problem.

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Bounded Duality in Topological Abelian Groups M. J. Chasco and E. Martín-Peinador

Dedicated to Juan Carlos Ferrando, celebrating his successful academic and research career

Abstract We define the β-duality for topological Abelian groups by means of the notion of Hejcman of boundedness in uniform spaces. A real locally convex space considered as an Abelian topological group is β-reflexive iff it is reflexive in the ordinary sense for locally convex spaces. Thus, β-reflexivity is the natural extension to Abelian topological groups of the well-known notion of reflexivity. We prove: 1) A locally compact Abelian group is β-reflexive. 2) A β-reflexive metrizable group is reflexive in Pontryagin sense. 3) The β-bidual of a metrizable group is also a metrizable group. Keywords H–bounded set · Reflexive · Equicontinuous · Precompact · Schwartz group · Locally convex space

1 Introduction Duality in locally convex spaces has been a flourishing topic in the second half of the 20th century. The monograph written by Grothendieck after a course explained in Sao Paulo in 1954 is a starting point for an abundant literature on this field. M. J. Chasco Departamento de Física y Matemática Aplicada, Universidad de Navarra, Pamplona, Spain e-mail: [email protected] E. Martín-Peinador (B) Instituto de Matemática Interdisciplinar y Departamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, Madrid, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_5

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The class of Abelian topological groups encloses the topological vector spaces as a distinguished subclass. In particular, the locally quasi convex Abelian groups generalize the locally convex spaces. Thus, notions of reflexivity and duality are natural also in the category of locally quasi-convex groups. The seminal work of Pontryagin in Locally Compact Abelian groups has been an inspiring point in considering duality for general Abelian topological groups. Kaplan, Varopoulos, Smith can be honoured as the initiators of this project. The cornerstone of their developments is to consider the compact-open topology in the character group of a topological group. In previous work we have either generalized or detected obstructions to extend important theorems on locally convex spaces, like Mackey Theorem, Grothendieck’s Completeness Theorem, Banach-Dieudonné Theorem, Eberlein-Šmulyan or Dunford-Pettis Theorem to the class of locally quasiconvex groups (See [5–8, 10]) . Also the definition of Schwartz spaces has given rise to that of Schwartz groups [3]. In the present paper, after considering the notion of boundedness modelled in the definition given by Hejcman for uniform spaces in [9], we start dealing with a new duality for Abelian topological groups. We have called it β-duality and in some sense is the most natural one in order to extend some results of Functional Analysis. In fact, we prove that a locally convex space E is reflexive iff it is β-reflexive considered as a topological group. We also prove that the β-bidual of a metrizable group is again metrizable. This duality opens a new field of research, and the present paper is only an introduction to it.

2 Notation, Definitions and Remarks Let us collect some facts and notation concerning group dualities. As usual, T denotes the multiplicative group of all complex numbers with modulus 1, with the topology induced by the Euclidean on C. We use the notation T+ = {t ∈ T : Re t ≥ 0}, where “Re” denotes the real part of a complex number. For an Abelian group G, Hom(G, T) denotes the group of all homomorphisms χ : G → T (also called characters), with multiplication defined pointwise. If G is a topological Abelian group, G ∧ will stand for the character group of G which is the subgroup of Hom(G, T) formed by all continuous characters. Let (G, H ) be a group duality (i. e. a pair formed by an Abelian group G and a subgroup H of Hom(G, T)). If H separates the points of G, we say that the duality is separating. The polar of a set A ⊂ G with respect to the duality (G, H ) is the set A = {χ ∈ H : χ(A) ⊂ T+ } The inverse polar of a set B ⊂ H with respect to the duality (G, H ) is the set

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B  = {x ∈ G : χ(x) ∈ T+ ∀χ ∈ B}. It is known ((1.4) in [4]) that any character in the polar of a zero neighborhood is continuous. A subset A of H is equicontinuous if there exists a zero neighborhood in G such that A ⊂ U  , or equivalently if A is a zero neighborhood in G (see [8]). A set A ⊂ G is said  to be quasi-convex with respect to the duality (G, H ) if A = A, that is, if A = χ∈A χ−1 (T+ ). If G is a topological Abelian group, quasiconvex subsets of G with respect to the duality (G, G ∧ ) will be said simply to be quasi-convex. A topological Abelian group is called locally quasi-convex if it has a basis of neighborhoods of zero formed by quasi-convex sets. This notion is originally due to Vilenkin, and later recovered in [4]. Locally quasi-convex groups play a similar role in the category of topological Abelian groups as locally convex spaces in that of topological vector spaces. In fact, it is well known (see [4]) that a topological vector space is locally convex if and only is it is locally quasi-convex as topological group. Given an Abelian group G and a symmetric subset U ⊂ G such that 0 ∈ U, we consider the following sequence of subsets of G: U(n) = {x ∈ G : x ∈ U, 2x ∈ U, . . . , nx ∈ U }, n ∈ N. It is natural to put U(∞) := {x ∈ G : nx ∈ U ∀n ∈ N}. Clearly 0 ∈ U(n) for every n ∈ N ∪ {∞}, and U(∞) ⊂ U(n+1) ⊂ U(n) for every n ∈ N. For a nonempty subset B ⊂ G and for a natural number n, n B will denote the set n

{nx : x ∈ B}. Obviously n B ⊂ B + B +....+ B. We now recall the notion of topology of uniform convergence on a family S or τS -topology, as it appears in [8]. A nonempty family S of subsets of G is called well-directed if the following conditions hold: (a) For B1 , B2 ∈ S, there exists B3 ∈ S such that B1 ∪ B2 ⊂ B3 . (b) For B ∈ S and n ∈ N, there exists A ∈ S, such that n B ⊂ A. If S is the family of all nonempty finite subsets, or of all compact subsets of G, S is well-directed. Let (G, H ) be a group duality and S a well-directed family of nonempty subsets of G. Since T is a metric space, we can consider in H ⊂ TG the topology τS (H, G) of uniform convergence on the sets A ∈ S. It will be called an S-topology, and it is a group topology. If S covers G, then τS (H, G) is Hausdorff. In the same fashion, if S is a well-directed family of nonempty subsets of H , and α : G → H om(H, T) the natural homomorphism, the preimage topology α−1 (τS (α(G), H )) will be denoted by τS (G, H ) and called the S -topology of G. Clearly a S -topology in G is a group topology and it is Hausdorff if H separates the points of G and S covers H . Let G be a group, H a subgroup of H om(G, T), S and S well-directed families of nonempty subsets of G and H respectively. The following facts hold:

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(a) The collection

B = {B  : B ∈ S }

is a fundamental system of neighborhoods of the neutral element eG in the topology τS (G, H ). In particular, τS (G, H ) is a locally quasi-convex topology. (b) The collection A = {A : A ∈ S} is a fundamental system of neighborhoods of the neutral element e H in the topology τS (H, G). In particular, τS (H, G) is locally quasi-convex. In order to introduce the β–duality, we first recall the notion of H–bounded subset (adapted from the definition given by Hejcman for uniform spaces). Let (G, τ ) be a topological group. A subset B of G is H–bounded if for every zero neighborhood U there exists a finite set F ⊂ G and some n ∈ N such that B ⊂ F + U + . n. . +U . The family of H–bounded sets of G is well-directed and the group G ∧ can be endowed with the topology τβ of uniform convergence on the H–bounded sets of G. Let us write G ∧β := (G, τβ )∧ . We will say that the group (G, τ ) is β-reflexive if the canonical evaluation mapping jG : G → (G ∧β )∧β defined by jG (x)(χ) = χ(x) is a topological isomorphism. The notion of H–bounded set is the natural extension to the class of topological groups of the well-known notion of bounded subsets in the context of topological vector spaces as indicated in the next statement, whose proof is easy. Lemma 2.1 If E is a locally convex vector space, A ⊂ E is bounded in E if and only if A is H–bounded in E as a topological Abelian group. If E is a real locally convex vector space we denote by E ∗ the dual vector space of continuous linear forms and by E β∗ the dual vector space E ∗ endowed with the topology of uniform convergence on the bounded subsets of E. Observe that the collection {B ◦ : B bounded subset of E} is a fundamental system of neighborhoods of zero for the topology τβ , where B ◦ = { f ∈ E ∗ : | f (x)| ≤ 1 for all x ∈ B}. As usual we say that E is reflexive if JE : E → (E β∗ )∗β defined by JE (x)( f ) = f (x) is a topological isomorphism. Let ρ be the universal covering map ρ : R → T given by ρ(s) = ex p(2πis). Proposition 2.2 Let E be a locally convex vector space, then ρ E : E β∗ → E β∧ given by ρ E ( f ) = ρ ◦ f is a topological isomorphism. Consequently, E is reflexive as a vector space iff E is β-reflexive as a topological group, Proof It was proved by Smith in [12] that ρ E : E ∗ → E ∧ is an algebraic isomorphism. The continuity of ρ E is clear because ρ E ((4B)◦ ) ⊂ B  for every subset B of

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E. In order to see that ρ E is open, take a neighborhood of zero U of E β∗ . There exists a bounded subset B of E such that B ◦ ⊂ 4U . The set H = {tu ∈ E : t ∈ [−1, 1], u ∈ B} is the balanced hull of B and therefore H is bounded (see [11, 5.1]); so H  is a neighborhood of zero in E β∧ . Moreover ρ E ((4H )◦ ) = H  because H is balanced (see [8, 1.11]). Thus H  = ρ E ( 41 H ◦ ) = ρ E ( 41 B ◦ ) ⊂ ρ E (U ) and ρ E (U ) is a neighborhood  of zero in E β∧ . Definition 2.3 ([3]) Let G be a Hausdorff topological Abelian group. We say that G is a Schwartz group if for every symmetric neighborhood of zero U in G there exists another neighborhood of zero V in G and a sequence (Fn ) of finite subsets of G such that V ⊂ Fn + U(n) for every n ∈ N. Remark 2.4 From the definition it follows directly that locally precompact Abelian groups (in particular, locally compact Abelian groups) are Schwartz groups. Observe that precompact subsets of an Abelian topological group are H–bounded. The converse holds for Schwartz groups, as proved in [3, 3.8]. We include below the proof for the reader’s convenience. Proposition 2.5 Let G be a locally quasi-convex Schwartz group and B an H– bounded subset of G. Then B is precompact. Proof Given a zero neighborhood U ⊂ G (which can be chosen quasi-convex), we must find a finite F ⊂ G such that B ⊂ F + U. Since G is a Schwartz group, there exist another neighborhood V ⊂ G and a sequence (Fn ) of finite subsets of G such that V ⊂ Fn + U(n) for every n ∈ N. On the other hand, by the H–boundedness of B there exist a finite set F0 ⊂ G and an m ∈ N such that B ⊂ F0 + V + . m. . + V . Hence B ⊂ F0 + V + . m. . + V ⊂ F0 + (Fm + U(m) ) + . m. . + (Fm + U(m) ) ⊂ F0 + (Fm + . m. . + Fm ) + (U(m) + . m. . + U(m) ) ⊂ F0 + (Fm + . m. . + Fm ) + U The fact that U(m) + · · · + U(m) ⊂ U derives from the quasi-convexity of U . Taking  F = F0 + (Fm + . m. . + Fm ), we have B ⊂ F + U. Proposition 2.6 ([3, 4.2]) Let E be a locally convex space. The following statements are equivalent: (a) E is a Schwartz space. (b) The additive topological Abelian group underlying E is a locally quasi-convex Schwartz group.

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Proof Among the known definitions of Schwartz spaces, we will use the following: a (real) locally convex space E is a Schwartz space if for every zero neighborhood U ⊂ E there exists another zero neighborhood V ⊂ E such that for every α > 0 there exists a finite subset Fα ⊂ E with V ⊂ Fα + αU. (a)⇒(b): This implication is quite trivial: without lost of generality, assume that U is a circled neighborhood of zero, specialize α = 1/n and observe that U(n) = 1/nU holds. (b)⇒(a): Fix an absolutely convex zero neighborhood U ⊂ E. There exists another neighborhood V and a sequence {Fn } of finite subsets of E such that V ⊂ Fn + U(n) for every n ∈ N. Let α > 0 and n ∈ N such that 1/n < α. Then V ⊂ Fn + U(n) ⊂ Fn + αU.  Proposition 2.7 ([3, 5.5]) Every Abelian topological group G which is a hemicompact k–space (in particular, every character group of a metrizable group), endowed with the topology of uniform convergence on compact sets is a Schwartz group. Theorem 2.8 Every locally compact Abelian group G is β - reflexive. Proof Every locally compact Abelian group G is locally quasi-convex and Schwartz, so by Proposition 2.5 the H–bounded sets of G are precompact, and by the completeness of G they are relatively compact. Therefore in the group of characters G ∧ , the compact-open topology coincides with the topology of uniform convergence on H–bounded sets. Thus, G ∧ β is again locally compact and consequently, the compactopen topology in (G ∧β )∧ coincides again with the topology of uniform convergence on H–bounded sets. In other words, (G ∧β )∧β is the Pontryagin bidual of G. As G is Pontryagin reflexive, it is also β-reflexive.  Proposition 2.9 If G is a topological Abelian group, then jβ : G −→ (G ∧β )∧β is continuous iff every H–bounded set of G ∧β is equicontinuous. Proof Suppose jβ is continuous and let B be an H–bounded subset of G ∧β . Then, B  is a zero neighborhood in (G ∧β )∧β . Since jβ is continuous, jβ−1 (B  ) = B  is a zero neighborhood in G and therefore B is equicontinuous. Conversely, take a zero neighborhood U in (G ∧β )∧β . There exists an H–bounded subset in G ∧β such that B  ⊂ U . From the equality jβ−1 (B  ) = B  , we have that jβ (B  ) ⊂ U . Thus, jβ is continuous.  Proposition 2.10 Let G be a locally quasi-convex topological Abelian group such that every equicontinuous subset of G ∧ is H–bounded in G ∧β . Then, jG : G −→ (G ∧β )∧β is open onto its image. Proof Let U be a quasi-convex zero neighborhood in G. Clearly U  is equicontinuous, and H–bounded by the assumption. Thus, (U  ) is a zero neighborhood in (G ∧β )∧β . Since jG (U ) = (U  ) ∩ jG (G), jβ is relatively open. 

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Remark 2.11 The equicontinuous subsets of the dual of a topological group may not be H–bounded with respect to τβ , as we explain right below. However, they are relatively compact with respect to the compact- open topology and therefore H–bounded in τco . In [9, 3.8] it is provided an example of a metrizable locally bounded group G, whose dual is not locally bounded with respect to the bounded convergence topology τβ . A locally bounded group is a topological group which has a bounded neighborhood of zero. Denote by V an H-bounded neighborhood of zero in the above mentioned example G. Clearly V  is an equicontinuous subset of G ∧ which is not H-bounded, for otherwise G ∧ should be locally bounded. Theorem 2.12 Every metrizable, β-reflexive topological Abelian group is Pontryagin reflexive. Proof Let G be a β-reflexive group, that is, jG : G −→ (G ∧β )∧β is a topological isomorphism. We must see that αG : G −→ (G ∧co )∧co is also a topological isomorphism. Observe that (G ∧β )∧β is locally quasi-convex, so G is locally quasi-convex which implies that αG : G −→ (G ∧co )∧co is relatively open and one to one. The continuity of αG follows from the assumption that G is metrizable and [1, 5.11] applies. The surjectivity of αG derives from the fact that τco ≤ τβ , and therefore (G ∧co )∧ ⊂ ∧ ∧  (G β ) , which by the hypothesis can be identified with G. Theorem 2.13 If G is a metrizable topological Abelian group, then jβ : G −→ (G ∧β )∧β is continuous and (G ∧β )∧β is also metrizable. Proof Let B ⊂ G ∧ be an H–bounded set in G ∧β . In particular, it is H–bounded in G ∧co and by Propositions 2.5 and 2.7 it is precompact in G ∧co . By the completeness of the latter, it is relatively compact and therefore equicontinuous. This proves that jG is continuous. Let {Vn , n ∈ N} be a decreasing basis of neighborhoods of zero in G. In order to see that (G ∧β )∧β is metrizable, we only need to prove that {Vn , n ∈ N} is a basis of neighborhoods of zero in (G ∧β )∧β . So let W be a zero neighborhood in (G ∧β )∧β . There exists an H-bounded set B of G ∧β such that B  ⊂ W . Since B is equicontinuous B  is a zero neighborhood in G, and consequently Vm ⊂ B  , for some some m ∈ N.  Thus, Vm ⊂ (B  ) = B  ⊂ W . The previous theorem uses a strong tool, namely the fact that the dual of a metrizable group is a Schwartz group. For topological vector spaces, a direct route to obtain a similar result can be described as follows. Theorem 2.14 If X is a metrizable topological vector space, then (X β∗ )∗β is also metrizable. If J : X −→ (X β∗ )∗β is onto, and X ∗ separates points of X , (X β∗ )∗β can be identified with the locally convex modification of X (via the algebraic isomorphism J ). First we prove some auxiliary results.

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∗ Proposition 2.15 If X is a metrizable topological vector space and X co its dual ∗ is endowed with the compact-open topology, then every bounded subset B ⊂ X co equicontinuous.

Proof Let {Vn , n ∈ N} be a decreasing basis of neighborhood of zero in X . Clearly { n1 Vn , n ∈ N} is also a decreasing basis of neighborhood of zero in X . Assume by contradiction that B is X c∗ -bounded but not equicontinuous. Then for all n ∈ N we can find f n ∈ B such that f n ( n1 Vn ) ⊂ [−1, 1]. Pick xn ∈ Vn such / [−1, 1]. Clearly xn −→ 0 by the election of {Vn , n ∈ N}. Thus that f n ( n1 xn ) ∈ S := {xn , n ∈ N} ∪ {0} is a compact subset of X and S ◦ a neighborhood of zero in / S ◦ for all n ∈ N and on the other hand B is bounded τco . On one hand we have n1 f n ∈ in X c∗ , which implies the existence of λ0 such that B ⊂ λS ◦ for all λ ≥ λ0 . If n ≥ λ0 , fn B ⊂ nS ◦ and thus ∈ S ◦ , a contradiction.  n Corollary 2.16 Let X be a metrizable topological vector space and B ⊂ X ∗ bounded in τβ . Then B is equicontinuous. Proof This derives from the fact that τco ≤ τβ and the previous proposition.



Lemma 2.17 Let X be a metrizable topological vector space. The canonical mapping J : X −→ (X β∗ )∗β is continuous. Proof Let L be a neighborhood of zero in (X β∗ )∗β . We may assume that L = B ◦ where B is a bounded subset of X β∗ . Clearly J −1 (B ◦ ) =◦ B, being ◦ B the inverse polar of B. From Corollary 2.16, B is equicontinuous, and therefore J −1 (B ◦ ) =◦ B is a neighborhood of zero in X .  Proposition 2.18 Let X be a topological vector space and V ⊂ X a neighborhood of zero. Then V ◦ is bounded in X β∗ . Proof We need to prove that for every bounded set B ⊂ X there is a real number r > 0 such that V ◦ ⊂ r B ◦ . Fix such a B. Boundedness of B implies that there exist  s > 0 such that B ⊂ sV . Taking polars we get (sV )◦ ⊂ B ◦ , thus V ◦ ⊂ s B ◦ . Proposition 2.19 Let X be a metrizable topological vector space. Then X β∗ has a fundamental sequence of bounded sets. Proof Let {Vn : n ∈ N} be a decreasing basis of zero neighborhood in X . Fix a bounded set B in X β∗ . By Corollary 2.16, it is equicontinuous. Thus, there exists n ∈ N such that B ⊂ Vn◦ . On the other hand, by Proposition 2.18, Vn◦ is bounded in X β∗ . Thus V1◦ ⊂ V2◦ ⊂ · · · ⊂ Vn◦ ⊂ · · · form a countable family of bounded subsets of X β∗ which is fundamental. The term “fundamental” is referred in the Literature also by “swallows bounded sets”. 

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Proof of Theorem 2.14. Since (X β∗ )∗β has a countable basis of neighborhoods, namely {(Vn◦ )◦ , n ∈ N}, it is metrizable. If J is onto, we can conclude that (X β∗ )∗β is the locally convex modification of (X, τ ). Acknowledgements Thanks are due to Xabier Domínguez for pointing out for us the example mentioned after Remark 2.11.

References 1. Außenhofer, L.: Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups. In: Dissertationes Mathematicae, vol. 384. Warsaw (1999) 2. Außenhofer, L.: A survey on nuclear groups. In: Nuclear Groups and Lie Groups. Proceedings of the Workshop on Topological Groups and Lie Groups, Madrid, 1999. Research and Exposition in Mathematics, vol. 24, pp. 1–30. Heldermann (2001) 3. Außenhofer, L., Chasco, M.J., Domínguez, X., Tarieladze, V.: On Schwartz groups. Studia Math. 181(3), 199–210 (2007) 4. Banaszczyk, W.: Additive subgroups of topological vector spaces. Lecture Notes in Mathematics, vol. 1466. Springer, Berlin-Heidelberg (1991) 5. Bruguera, M., Martín-Peinador, E.: Banach-Dieudonné theorem revisited. J. Aust. Math. Soc. 75, 1–15 (2003) 6. Bruguera, M., Martín-Peinador, E., Tarieladze, V.: Eberlein-Šmulyan Theorem for abelian topological groups. J. Lond. Math. Soc. 70(2), 341–355 (2004) 7. Bruguera, M., Chasco, M.J., Martín-Peinador, E., Tarieladze, V.: Completeness properties of locally quasi-convex groups. Topology Appl. 111, 81–95 (2001) 8. Chasco, M.J., Martín-Peinador, E., Tarieladze, V.: On Mackey topology for groups. Stud. Math. 132(3), 257–284 (1999) 9. Hejcman, J.: Boundedness in uniform spaces and topological groups. Czechoslovak Math. J. 9(84), 544–562 (1962) 10. Martín-Peinador, E., Tarieladze, V.: A property of Dunford-Pettis type in topological groups. PAMS 132(6), 1827–1837 (2003) 11. Schaefer, H.: Topological vector spaces. In: Graduate Texts in Mathematics, vol. 3 (1970) 12. Smith, M.F.: The Pontryagin duality theorem in linear spaces. Ann. Math. 56(2), 248–253 (1952)

Topological Properties of the Weak and Weak∗ Topologies of Function Spaces Saak Gabriyelyan

In Honour of Juan Carlos Ferrando, Loyal Friend and Excellent Mathematician

Abstract We show that if a locally convex space (lcs) (E, τ ) is a sequentially Ascoli space in the weak topology, then τ = σ (E, E  ). Consequently, a complete lcs E is weakly sequentially Ascoli iff E = Fλ for some cardinal λ. For a Tychonoff space X , let Ck (X ) be the space C(X ) of all continuous functions on X endowed with the compact-open topology. We prove that: (1) Ck (X ) is weakly sequentially Ascoli iff Ck (X ) is weakly Ascoli iff it is weakly κ-Fréchet–Urysohn iff X has the property (κ) and every compact subset of X is finite; (2) Ck (X ) is weakly Fréchet–Urysohn iff Ck (X ) is a weakly sequential space iff it is a weakly k-space iff the space X has the property γ whose compact subsets are finite; (3) the dual space of Ck (X ) is weak∗ sequentially Ascoli iff X is finite. If Cb (X ) is the space C(X ) with the topology of uniform convergence on functionally bounded subsets of X , then: (a) Cb (X ) is weakly sequentially Ascoli iff Cb (X ) is weakly Ascoli iff it is weakly κFréchet–Urysohn iff X has the property (κ) and every functionally bounded subset of X is finite; (2) Cb (X ) is weakly Fréchet–Urysohn iff Cb (X ) is a weakly sequential space iff it is a weakly k-space iff the space X has the property γ whose functionally bounded subsets are finite. Keywords Weak topology · Weak∗ topology · Function space · Sequentially Ascoli space · Sequential space

S. Gabriyelyan (B) Ben-Gurion University of the Negev, P.O. 653 Beer Sheva, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_6

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1 Introduction A systematic study of topological properties of Banach spaces endowed with the weak topology was proposed by Corson [5]. It is natural to consider a more general problem: Study topological properties of a locally convex space endowed with the weak topology. All locally convex spaces (lcs for short) are over the field F of real or complex numbers. For a Tychonoff space X , we denote by Ck (X ) and C p (X ) the space C(X ) of all (real or complex) continuous functions on X endowed with the compact-open topology or the pointwise topology, respectively. Now we recall the most important topological properties considered in the article which generalize metrizability. A Tychonoff space X is called • Fréchet–Urysohn if for any cluster point a ∈ X of a subset A ⊆ X there is a sequence {an }n∈N ⊆ A which converges to a; • κ-Fréchet–Urysohn if for every open subset U of X and every x ∈ U , there exists a sequence {xn }n∈ω ⊆ U converging to x; • sequential if for each non-closed subset A ⊆ X there is a sequence {an }n∈N ⊆ A converging to some point a ∈ A¯ \ A; • a k-space if for each non-closed subset A ⊆ X there is a compact subset K ⊆ X such that A ∩ K is not closed in K ; • a kR -space if a real-valued function f on X is continuous if and only if its restriction f  K to any compact subset K of X is continuous; • Ascoli if every compact subset K of Ck (X ) is evenly continuous, that is the map X × K  (x, f ) → f (x) ∈ R is continuous; in other words, X is Ascoli if and only if the compact-open topology of Ck (X ) is Ascoli in the sense of [19, p. 45]; • sequentially Ascoli if every convergent sequence in Ck (X ) is equicontinuous. The notion of a κ-Fréchet–Urysohn space was introduced by Arhangel’skii. Ascoli spaces were introduced in [2] being motivated by the classical Ascoli theorem which states that if X is a Tychonoff k-space, then every compact subset K of Ck (X ) is evenly continuous (see Theorem 3.4.20 in [6]). The notion of a sequentially Ascoli space was introduced in [11] and studied in [13]. The following diagram describes the relationships between the aforementioned properties:  Ascoli

κ-Fréchet– Urysohn

metric

 Fréchet– Urysohn

 sequential

 k-space

 sequentially Ascoli

 kR -space

None of these implications is reversible, see [2, 6, 10, 11, 14]. Below we summarize the most interesting and important results some of which, namely (viii)–(xi), we shall use to prove the main results of the article, see Theorems 3.3 and 3.8.

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Theorem 1.1 (i) ([23]) An infinite-dimensional Banach space in the weak topology is never a k-space. (ii) ([16]) A Banach space E is weakly Ascoli if and only if it is finite-dimensional. (iii) ([7]) A Fréchet space E is weakly Ascoli if and only if E = Fn or E = Fω . (iv) ([1, 16]) The closed unit ball of a Banach space E endowed with the weak topology is an Ascoli space if and only if E does not contain an isomorphic copy of 1 . (v) ([9]) A strict (L F)-space E is a sequentially Ascoli space if and only if E is a Fréchet space or E = ϕ. (vi) ([9]) The space of distributions D  ( ) is not a sequentially Ascoli space. (viii) (Gerlits–Nagy–Pytkeev Theorem) C p (X ) is Fréchet–Urysohn if and only if it is a k-space if and only if X has the property γ . (ix) ([22]) C p (X ) is κ-Fréchet–Urysohn if and only if X has the property (κ). (x) ([10, 15]) C p (X ) is an Ascoli space if and only if X has the property (κ). (xi) ([13]) C p (X ) is a sequentially Ascoli space if and only if X has the property (κ). Now we describe the content of the paper. In Theorem 1.6 of [7] we proved that if a c0 -barrelled space (E, τ ) is weakly Ascoli, then the topology τ of E coincides with the weak topology σ (E, E  ). In Sect. 2 we show that the condition on E to be c0 -barrelled can be omitted, see Theorem 2.1. This result gives a necessary condition of being a weakly sequentially Ascoli space. As a consequence we show in Corollary 2.2 that a complete lcs E is weakly sequentially Ascoli if and only if E = Fλ for some cardinal λ. In Sect. 3 we characterize Tychonoff spaces X for which the space CT (X ) endowed with one of the most important “set-open” topologies T (for example the compact-open topology) is a weakly sequentially Ascoli space. In particular, the space Ck (X ) is weakly sequentially Ascoli if and only if it is weakly κ-Fréchet– Urysohn if and only if X has the property (κ) and every compact subset of X is finite, see Theorem 3.3. It turns out, see Theorem 3.8, that Ck (X ) is weakly Fréchet– Urysohn if and only if and only if it is weakly k-space if and only if the space X has the property γ whose compact subsets are finite. To prove both Theorems 3.3 and 3.8 we essentially use (viii)–(xi) of Theorem 1.1 and Theorem 2.1. In [7] we proved that if X is a μ-space, then the dual space of Ck (X ) is weakly∗ Ascoli if and only if X is finite. In the main result of Sect. 4, Theorem 4.4, we considerably generalize this result by showing that the dual space of Ck (X ) is weakly∗ sequentially Ascoli if and only if X is finite.

2 A Necessary Condition of Being a Weakly Sequentially Ascoli Space For an lcs E we shall use the following standard notations: E  denotes the topological dual space of E, E w := E, σ (E, E  ) and the dual space E  endowed with the strong  the dual space E  endowed with topology β(E  , E) is denoted by E β . Denote by E wc

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the topology Twc of uniform convergence on weakly compact subsets of E. Since every weakly compact set of E is bounded, it follows that Twc ⊆ β(E  , E). Theorem 2.1 Let (E, τ ) be an lcs such that E w is a sequentially Ascoli space. Then: (i) τ = σ (E, E  ), so E is linearly isomorphic to a dense subspace of Fλ for some cardinal λ;  and hence of E β is finite-dimensional. (ii) every bounded subset of E wc Proof (i) Suppose for a contradiction that σ (E, E  )  τ . Fix a closed absolutely convex τ -neighborhood U of zero in E such that U ∈ / σ (E, E  ). Then the polar ◦ ∗ K := U of U is a weak compact absolutely convex subset of E  by the Alaoglu Theorem, and hence, by Theorem 11.11.5 of [20], K is bounded in the strong topology β(E  , E) on E  . Note also that K is not finite-dimensional since, otherwise, U = K ◦ would belong to σ (E, E  ). Fix any linearly independent sequence {χn }n∈N in K . Then the sequence S := { n1 χn }n∈N is β(E  , E)-null. We claim that S is a null sequence also in Ck (E w ). To this end, let C be an arbitrary weakly compact subset of E. Then C is bounded in E, and hence C ◦ is a β(E  , E)neighbourhood of zero in E  . Therefore n1 χn ∈ C ◦ for all sufficiently large n ∈ N. Thus n1 χn → 0 in the space Ck (E w ). By assumption, E w is a sequentially Ascoli space. Therefore the sequence S is equicontinuous, and hence there are a finite subset F of E  and ε > 0 such that S ⊆ [F; ε]◦ where [F; ε] := {x ∈ E : |χ (x)| ≤ ε for all χ ∈ F}. Consequently S is contained in the finite-dimensional subspace span(F) of E  , and hence S is linearly dependent, a contradiction.  which (ii) Suppose for a contradiction that there is a bounded subset A of E wc is infinite-dimensional. We can assume that A = {χn }n∈N . Then the sequence S := { n1 χn }n∈N converges to zero in Twc . Hence S is a null sequence also in Ck (E w ). Now we repeat the last paragraph of the proof of (i) to get a contradiction.  The next corollary essentially generalizes Corollary 1.7 of [7]. Corollary 2.2 A complete lcs E is weakly sequentially Ascoli if and only if E = Fλ for some cardinal λ. Proof By Theorem 2.1, the space E is a dense subspace of Fλ for some cardinal λ. Being also complete E coincides with Fλ . Conversely, if E = Fλ for some cardinal λ, then E is even a κ-Fréchet–Urysohn space, see for example Corollary 2.3 of [10].  The following immediate consequence of Corollary 2.2 generalizes (i)–(iii) of Theorem 1.1. Corollary 2.3 A Fréchet space E is weakly sequentially Ascoli if and only if E = F N for some N ≤ ℵ0 .

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For an lcs E, we denote by E w ∗ the dual space E  endowed with the weak∗ topology σ (E  , E). The following problem arises naturally: Characterize E whose weak∗ dual space E w ∗ is a sequentially Ascoli space. By definition this means that every null sequence in Ck (E w ∗ ) is equicontinuous. As every x ∈ E belongs to Ck (E w ∗ ), one can ask: When the space E is a subspace of Ck (E w ∗ )? If the space E is barrelled then E is a subspace of Ck (E w ∗ ) by Lemma 4.8 of [7]. To obtain a complete characterization we recall that an lcs E is called g-barrelled (c0 -barrelled) if any weak∗ compact subset (resp. weak∗ null sequence) of E  is equicontinuous. The notion of g-barrelledness was introduced in [4] for abelian topological groups. Clearly, every barrelled space is g-barrelled and each g-barrelled space is c0 -barrelled, but the converse is not true in general. An example of a g-barrelled lcs which is not barrelled has pointed out in Remark 16 of [4], for another example with a detailed proof see Example 5.6 in [8]. Below we provide a useful functional characterization of g-barrelled and c0 -barrelled spaces. For a Tychonoff space X , we denote by Cs (X ) the space C(X ) endowed with the topology of uniform convergence on finite unions of convergent sequences in X . Proposition 2.4 (i) An lcs E is g-barrelled if and only if the canonical map i : E → Ck (E w ∗ ), (i(x), χ ) := χ (x) for χ ∈ E  , is continuous. In this case i is an embedding, and i(E) is a closed subspace of Ck (E w ∗ ). (ii) An lcs E is c0 -barrelled if and only if the canonical map i : E → Cs (E w ∗ ), (i(x), χ ) := χ (x) for χ ∈ E  , is continuous. Proof We shall prove only (i) because clause (ii) can be proved analogously. Assume that E is g-barrelled and let [K ; ε] = { f ∈ C(E w ∗ ) : | f (x)| ≤ ε for all x ∈ K } be a standard neighborhood of zero in Ck (E w ∗ ), where K is compact subset of E w ∗ and ε > 0. Since E is g-barrelled, there is a neighborhood U of zero in E such that 1 K ⊆ U ◦ . Then for every χ ∈ K and each x ∈ U , we have |(i(x), χ )| = |χ (x)| ≤ ε ε and hence i(U ) ⊆ [K ; ε]. Thus i is continuous. Conversely, assume that i is continuous. Fix a weak∗ compact subset K of E  . Then U := i −1 [K ; 1] is a neighborhood of zero in E. Therefore, for every χ ∈ K and each x ∈ U , we have |χ (x)| = |(i(x), χ )| ≤ 1 and hence K ⊆ U ◦ . Thus K is equicontinuous and hence E is g-barrelled. To show that i is an embedding, it suffices to show that i is an open map onto its image. To this end, fix an arbitrary closed absolutely convex neighborhood U of zero in E, then U ◦ is a compact subset of E w ∗ (by Alaoglu’s theorem) and i(U ) = i(U ◦◦ ) = {x ∈ E : |χ (x)| ≤ 1 for all χ ∈ U ◦ } = i(E) ∩ [U ◦ ; 1]. Thus i is open. To show that i(E) is closed, fix a continuous function f ∈ Ck (E w ∗ ) which is a cluster point of i(E). It is clear that f is linear and hence f ∈ (E w ∗ ) = E. Thus f ∈ i(E). 

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Recall that an lcs E is called feral if every bounded set B ⊆ E is finitedimensional. If E is feral, then every weakly null sequence S in E is finitedimensional and hence equicontinuous with respect to E w ∗ . Thus E w ∗ is c0 -barrelled. If E is in addition c0 -barrelled, the converse assertion is also true. Proposition 2.5 Let E be a c0 -barrelled lcs. Then E w ∗ is c0 -barrelled if and only if E is feral. Proof Assume that E w ∗ is c0 -barrelled and suppose for a contradiction that E is not feral. Then E has a linearly independent bounded sequence S = {xn }n∈ω . Replacing xn by n1 xn we can assume that xn → 0 in E, and hence S is also weakly null. Since E w ∗ is c0 -barrelled, S is equicontinuous with respect to E w ∗ . So there is a finite subset F of E such that S ⊆ F ◦◦ . Since F is finite, the set F ◦◦ is finite-dimensional. Therefore also S is finite-dimensional, a contradiction. Conversely, assume that E is feral. Then every weakly null sequence S in E is finite-dimensional, and being bounded in E it is equicontinuous. Thus E w ∗ is  c0 -barrelled. Theorem 1.12 of [7] states that if E is barrelled and E w ∗ is an Ascoli space, then E is feral. Below we generalize this result. Theorem 2.6 Let E be a c0 -barrelled lcs. If E w ∗ is a sequentially Ascoli space, then E is feral and E w ∗ is c0 -barrelled. Proof Since E is c0 -barrelled, Proposition 2.4 implies that the canonical injective map i : E → Ck (E w ∗ ) is continuous. As E w ∗ is sequentially Ascoli, each null sequence in Ck (E w ∗ ) is equicontinuous. Now suppose for a contradiction that E is not feral. Then E has a linearly independent bounded sequence S = {xn }n∈ω . Replacing xn by n1 xn we can assume that xn → 0 in E, and hence i(S) is equicontinuous with respect to E w ∗ . So there is a finite subset F of E such that i(S) ⊆ F ◦◦ . Since F is finite, the set F ◦◦ is finite-dimensional. Therefore also S is finite-dimensional and hence linearly dependent, a contradiction. Finally, the space E w ∗ is c0 -barrelled by Proposition 2.5. 

3 Function Spaces Which are Weakly Sequentially Ascoli We start from some necessary notations. Let X be a set, and let f : X → F be a function to the field F of real or complex numbers. For a subset A ⊆ X and ε > 0, let

f A := sup({| f (x)| : x ∈ A} ∪ {0}) ∈ [0, ∞], and if F is a subfamily of F X , we set [A; ε]F := { f ∈ F : f A ≤ ε}.

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If the family F is clear from the context, then we shall omit the subscript F and write [A; ε] instead of [A; ε]F . A family S of subsets of X is directed if for any sets A, B ∈ S the union A ∪ B is contained in some set C ∈ S . For a Tychonoff space X , we denote by C(X ) the space of all continuous functions f : X → F on X . A subset A ⊆ X is called functionally bounded if f A < ∞ for any continuous function f ∈ C(X ). A Tychonoff space X is pseudocompact if X is functionally bounded in X . For a Tychonoff space X , the space C(X ) carries many important locally convex topologies, i.e., topologies turning C(X ) into a locally convex space. For a locally convex topology T on C(X ), we denote by CT (X ) the space C(X ) endowed with the topology T . Each directed family S of functionally bounded sets in a Tychonoff space X induces a locally convex topology TS on C(X ) whose neighborhood base at zero consists of the sets [S; ε] where S ∈ S and ε > 0. The topology TS is called the topology of uniform convergence  on sets of the family S . The topology TS is Hausdorff if and only if the union S is dense in X . If S is the family F (X ) of all finite (resp. compact K (X ) or functionally bounded F B(X )) subsets of X , then the topology TS will be denoted by T p (resp. Tk and Tb ), and the function space CT S (X ) will be denoted by C p (X ) (resp. Ck (X ) or Cb (X )). To prove the main result of this section we shall use the following two assertions. The next one is Lemma 4.6 from [7]. Lemma 3.1 ([7]) Let {yn }n∈ω be an independent sequence in a locally convex space E. Then for every finite subset {z 0 , . . . , z m } of E  there are a0 , . . . , am+1 ∈ F such that m  ker(z i ). 0 = a0 y0 + · · · + am+1 ym+1 ∈ i=0

For the case when S = K (X ) is the set of all compact subsets of X , the following assertion is Proposition 4.7 of [7]. Proposition 3.2 Let X be a Tychonoff space, and let S be a directed family of functionally bounded subsets of X containing F (X ). Then the topology TS on C(X ) coincides with the weak topology τw of CT S (X ) if and only if S = F (X ). In this case TS = T p = τw . Proof Let TS = τw and suppose for a contradiction that there is an infinite set K ∈ S . To get a contradiction it is sufficient to show that the TS -neighborhood [K ; 1] of 0 ∈ C(X ) does not contain a τw -neighborhood of zero. Since K is infinite, there is an infinite discrete sequence {xn }n∈ω in K with pairwise disjoint neighborhoods Vn of xn in X , see [17, Lemma 11.7.1]. For every n ∈ ω, choose a continuous function f n : X → [0, 1] with support in Vn and f n (xn ) = 1. Clearly, the functions f n are linearly independent. Now let U be an arbitrary weak neighborhood of zero in CT S (X ). Without loss of generality can assume that there are ε > 0 and a finite family  we  {μ0 , . . . , μm } ∈ CT S (X ) such that

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U = {g ∈ C(X ) : |μi (g)| < ε for all i = 0, . . . , m}. By Lemma 3.1, there are a0 , . . . , am+1 ∈ F such that 0 = h := a0 f 0 + · · · + am+1 f m+1 ∈

m 

ker(μi ).

(1)

i=0

Since h(xi ) = ai f i (xi ) = ai for every i = 0, . . . , m + 1, we obtain that h K > 0. It follows from (1) that h 2 K h ∈ U . On the other hand, since {Vn }n∈ω is disjoint and \Vn ) =  {0} there is j ∈ {0, . . . , m + 1} such that |h(x j )| = h K and hence f n (X  2 h(x j ) = 2 > 1. As x j ∈ K we obtain 2 h ∈ / [K ; 1], and hence U  [K ; 1].

h K

h K Consequently TS is strictly finer than τw , a contradiction. Thus every S ∈ S is finite and hence S = F (X ). Conversely, if S = F (X ), then the equalities TS = T p = τw hold trivially.  A family {Ai }i∈I of subsets of a set X is called point-finite if for every x ∈ X , the set {i ∈ I : x ∈ Ai } is finite. A family {Ai }i∈I of subsets of a topological space X is said to be strongly point-finite if for every i ∈ I there is an open set Ui of X such that Ai ⊆ Ui and the family {Ui : i ∈ I } is point-finite. Following Sakai [22], a topological space X is said to have the property (κ) if every pairwise disjoint sequence of finite subsets of X has a strongly point-finite subsequence. The following theorem gives in particular a complete answer to Problem 1.10 from [7]. Theorem 3.3 Let X be a Tychonoff space, and let S be a directed family of functionally bounded subsets of X containing F (X ). Then the following assertions are equivalent: (i) (ii) (iii) (iv)

CT S (X ) is a weakly κ-Fréchet–Urysohn space; CT S (X ) is a weakly Ascoli space; CT S (X ) is a weakly sequentially Ascoli space; X has the property (κ) and S = F (X ).

Proof The implications (i)⇒(ii)⇒(iii) are clear, see the diagram. (iii)⇒(iv) Assume that CT S (X ) is weakly sequentially Ascoli. Then, by Theorem 2.1, the topology TS coincides with the weak topology τw . Therefore, by Proposition 3.2, S = F (X ) and hence CT S (X ) = C p (X ). Therefore C p (X ) is a sequentially Ascoli space. Thus, by (xi) of Theorem 1.1, X has the property (κ). (iv)⇒(i) Assume that X has the property (κ) and S = F (X ). Then CT S (X ) = C p (X ). By the Sakai Theorem 1.1(ix), the space C p (X ) is κ-Fréchet–Urysohn. It  remains to note that C p (X ) carries its weak topology. The next corollary generalizes Corollary 1.9 of [7] in particular by removing the condition on X of being a μ-space.

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Corollary 3.4 If X is a kR -space, then Ck (X ) is a weakly sequentially Ascoli space if and only if X is discrete. Proof Assume that Ck (X ) is a weakly sequentially Ascoli space. Then, by Theorem 3.3, Ck (X ) = C p (X ) and hence it is a dense subspace of F X . Since X is a kR -space, the space Ck (X ) is complete by Corollary 5.1.2 of [19], and hence Ck (X ) = F X . Thus X is discrete. Conversely, if X is discrete, then Ck (X ) = F X is even a κ-Fréchet– Urysohn space, see for example Corollary 2.3 of [10].  It is known (and easy to show) that Ck (X ) = Cb (X ) if and only if X is a μ-space. This remark and Theorem 3.3 motivate the nex problem. Problem 3.5 Does there exist a pseudocompact non-compact space X having the property (κ) and whose compact subsets are finite? Theorem 3.6 Let X be a Tychonoff space, and let S be a directed family of functionally bounded subsets of X containing F (X ). Then CT S (X ) is a weakly kR -space if and only if S = F (X ) and C p (X ) is a kR -space. Proof Assume that CT S (X ) is a weakly kR -space. Then, by Theorem 2.1, the topology TS coincides with the weak topology τw . Therefore, by Proposition 3.2, S = F (X ) and CT S (X ) = C p (X ), and hence C p (X ) is a kR -space. Conversely, assume that S = F (X ) and C p (X ) is a kR -space. Then CT S (X ) = C p (X ), and since C p (X ) carries its weak topology, the space CT S (X ) is a weakly kR -space.  Theorem 3.6 motivates the following problem. Problem 3.7 Characterize Tychonoff spaces X for which C p (X ) is a kR -space. Let X be an arbitrary set. A sequence γ = {Un }n∈ω of subsets of X is called a   p-sequence if X = n∈ω i≥n Ui . A family γ of subsets of X is called an ω-cover of X if for any finite A ⊆ X , there is U ∈ γ such that A ⊆ U . Recall that a Tychonoff space X is said to have the property γ if every open ω-cover contains a p-sequence. Now we characterize spaces X for which CT S (X ) in the weak topology is Fréchet– Urysohn or a k-space. Theorem 3.8 Let X be a Tychonoff space, and let S be a directed family of functionally bounded subsets of X containing F (X ). Then the following assertions are equivalent: (i) (ii) (iii) (iv)

CT S (X ) is a weakly Fréchet–Urysohn space; CT S (X ) is a weakly sequential space; CT S (X ) is a weakly k-space; X has the property γ and S = F (X ).

Proof The implications (i)⇒(ii)⇒(iii) are clear, see the diagram. (iii)⇒(iv) Assume that CT S (X ) is a weakly k-space. Then, by Theorem 2.1, the topology TS coincides with the weak topology τw . Therefore, by Proposition 3.2,

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S = F (X ) and hence CT S (X ) = C p (X ). Therefore C p (X ) is a k-space. Thus, by (viii) of Theorem 1.1, X has the property γ . (iv)⇒(i) Assume that X has the property γ and S = F (X ). Then CT S (X ) = C p (X ). By the Gerlits–Nagy–Pytkeev Theorem 1.1(viii), the space C p (X ) is Fréchet–  Urysohn. Since C p (X ) carries its weak topology the proof is finished.

4 Free Locally Convex Spaces and the Space of Measures with Compact Support One of the most important classes of locally convex spaces is the class of free locally convex spaces introduced by Markov in [18]. The free locally convex space L(X ) over a Tychonoff space X is a pair consisting of a locally convex space L(X ) and a continuous map i : X → L(X ) such that every continuous map f from X to a locally convex space E gives rise to a unique continuous linear operator  E ( f ) : L(X ) → E with f =  E ( f ) ◦ i. The free locally convex space L(X ) always exists and is essentially unique. In what follows we shall consider i(x) as the Dirac measure δx at the point x ∈ X . We also recall that C p (X ) = L(X ) and L(X ) = C(X ). In [12] we proved that L(X ) is an Ascoli space if and only if X is a countable discrete space. Proposition 4.1 For every infinite Tychonoff space X , the free lcs L(X ) is not a weakly sequentially Ascoli space. Proof By Proposition 3.11 of [3], the space L(X ) does not carry its weak topology and Theorem 2.1 applies.  Recall that a family F ⊆ C(X ) is called pointwise bounded if the set { f (x) : f ∈ F } is bounded for every x ∈ X ; and F is equicontinuous if for every x ∈ X and each ε > 0 there is a neighborhood Ux of x such that | f (x  ) − f (x)| < ε for each x  ∈ Ux . By [21], the topology of the free lcs L(X ) is the topology of uniform convergence on equicontinuous pointwise bounded subsets of C(X ). The next two lemmas are known, but hard to find explicitly stated. So we add their detailed proofs to make the paper self-contained and for the convenience of the reader. Lemma 4.2 If X is a dense subspace of Z , then L(X ) is a dense subset of L(Z ). Proof It suffices to check that for every z ∈ Z and each standard neighborhood [F ; ε] of zero in L(Z ) there is x ∈ X such that δx ∈ δz + [F ; ε]. Since F is equicontinuous at z, there is a neighborhood U of z such that | f (z  ) − f (z)| < ε for each z  ∈ U.

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Since X is dense in Z , there is x ∈ U ∩ X . Then for every f ∈ F we obtain |δx ( f ) − δz ( f )| = | f (x) − f (z)| < ε, and hence δx ∈ δz + [F ; ε], as desired.



For a Tychonoff space X , we denote by Mc (X ) the space of all Borel regular measures on X with compact support. It is well known (see [17]) that Ck (X ) = Mc (X ). Denote by τe the topology on Mc (X ) of uniform convergence on all pointwise bounded equicontinuous subsets of C(X ). Lemma 4.3 For every Tychonoff space X , the free lcs L(X ) is a dense subspace of (Mc (X ), τe ). Proof Fix μ ∈ Mc (X ), and let K := supp(μ) be the compact support of μ. The Hahn decomposition theorem states that μ = μ+ − μ− , where μ+ , μ− ∈ Mc (X ) are non-negative measures. Therefore it suffices to prove the lemma assuming that μ is positive. Fix a standard τe -neighborhood [F ; ε] of zero, where ε > 0 and F ⊆ C(X ) is pointwise bounded and equicontinuous. We have to find χ ∈ L(X ) such that χ ∈ μ + [F; ε]. For every x ∈ K , choose an open neighborhood Ux of x such that | f (x  ) − f (x)|
0 : B covered by a finite number of balls of radii ≤  , respectively, for each B ∈ B X . It is immediate that Definition 2.1 implies the next result.

(2)

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Lemma 2.1 Let (X, ·) be a Banach space and B X the class of all nonempty bounded subsets of X . Then κ, χ : B X −→ R+ defined in (1) and (2) satisfy the above conditions (i)-(v) of Definition 2.1. It is easy to check that if (X, ·) is an infinite dimensional Banach space and B X its closed unit ball, then κ(B X ) ≤ 2. But Furi and Vignoli [18] and Nussbaum [41] proved that κ(B X ) = 2. The Kuratowski and Hausdorff MNCs are related by means of the following inequalities (see [5, Remark I.3.2]) χ (B) ≤ κ(B) ≤ 2χ (B), for all B ∈ B X .

(3)

Furthermore, the above inequalities can be improved in special Banach spaces such as Hilbert spaces or l p spaces with 1 ≤ p < ∞ (see [3, 8]). In the first case the inequalities (3) become √ 2χ (B) ≤ κ(B) ≤ 2χ (B), for all B ∈ B X , whereas in the second one we have √ p 2χ (B) ≤ κ(B) ≤ 2χ (B), for all B ∈ B X . We have just covered the basics of MNCs, although its development has been enormous. To complete the information on this theory, we suggest consulting the texts [3, 5, 7, 8, 40]. As we said, the theory of α-dense curves appeared in 1997 but the notion of DND arose in 2015 as an application of such a theory. Hence it is convenient to start with α-dense curves. Definition 2.2 Given α ≥ 0 and B ∈ B X , a continuous mapping γ : I −→ X is said to be an α-dense curve in B if: 1. γ (I ) ⊂ B. 2. for any x ∈ B there exists t ∈ I such that x − γ (t) ≤ α. Definition 2.3 A set B ∈ B X is said to be densifiable if there exists an α-dense curve in B for every α > 0. Given α ≥ 0 and B ∈ B X , consider the family   α,B := curves α-denses in B . Given B ∈ B X , let us note that α,B could be empty for some values of α. However, if we denote the diameter of B as diam(B), for any α ≥ diam(B), the family α,B = ∅

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because for an arbitrary x0 ∈ B, the curve γ : I → X defined as γ (t) := x0 for all t ∈ I is clearly α-dense in B. Then γ ∈ α,B , so α,B = ∅. This observation allows us to introduce the concept of DND of a set B ∈ B X . Definition 2.4 Given B ∈ B X , we define the DND of B and denote it by (B) as   (B) := inf α ≥ 0 : α,B = ∅ .

(4)

An immediate consequence from above definition is the next result. Proposition 2.1 Let (X, ·) be a Banach space and B X the class of all nonempty bounded subsets of X . If B ∈ B X is densifiable, then (B) = 0. Proof If B ∈ B X is densifiable, then for any α > 0, there exists an α-dense curve in B. It means that the class α,B = ∅ for arbitrary α > 0. Hence, from (4), it follows (B) = 0. Now we introduce the subfamily Ba X ⊂ B X defined by the class of all arcconnected sets of B X . In the class Ba X we can give the following characterization of densifiable sets, see [39, Proposition 2]. Proposition 2.2 A set B ∈ Ba X is densifiable if and only if B is precompact. The DND of the closed unit ball of a Banach space is an indicator of the dimension of that space, see [38]. Theorem 2.1 Let (X, ·) be a Banach space, B X its closed unit ball and  the DND defined on the family B X . Then  (B X ) =

0, if X is finite dimensional . 1, if X is infinite dimensional

A first relation of DND with Kuratowski MNC is the following [22, Theorem 2.7]: Theorem 2.2 Let (X, ·) be a Banach space and B X the class of all nonempty bounded subsets of X . Then the Kuratowski MNC and the DND are related by κ(B) ≤ 2(B), for all B ∈ B X .

(5)

Furthermore, the above inequality is the best possible in infinite dimensional Banach spaces. In the next result is related the DND with the Hausdorff MNC [22, Theorem 2.3].

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Theorem 2.3 Let (X, ·) be a Banach space and B X the class of all nonempty bounded subsets of X . Then the Hausdorff MNC and the DND are related in the following way (6) χ (B) ≤ (B), for all B ∈ B X . Furthermore, the above inequality is the best possible in infinite dimensional Banach spaces. In the subfamily Ba X we obtain a complete relation of the Hausdorff MNC with the DND [22, Theorem 2.5]. Theorem 2.4 Let (X, ·) be a Banach space and Ba X the subclass of all arcconnected sets of B X . Then the Hausdorff MNC and the DND satisfy χ (B) ≤ (B) ≤ 2χ (B), for all B ∈ Ba X .

(7)

Furthermore, the above inequality is the best possible in infinite dimensional Banach spaces. It is important to stress that if we substitute the subclass Ba X by the class B X , the second inequality in (7) is not true even in finite dimensional Banach spaces, as it follows from the next example. Likewise this example proves that the DND is not a MNC. Example The topologist’s sine curve (see for instance [30]) defined as     Sc := (x, sin(1/x)) : x ∈ [−1, 0) ∪ (0, 1] ∪ (0, y) : y ∈ [−1, 1] satisfies χ (Sc ) = 0 and (Sc ) ≥ 1. Hence in (7) the second inequality is only true for nonempty, bounded and arc-connected sets of (X, ·). Furthermore,  is not a MNC because it does not fulfill the regularity property of definition of MNC. The topologist’s sine curve Sc also points out that only the second inequality in (B) ≤ κ(B) ≤ 2(B), for all B ∈ B X ,

(8)

is true, as we have just seen in (5). However the first inequality in (8) is false in general. Indeed, since Sc is closed and bounded, Sc is a compact set in X := R2 for the usual topology, so obviously we have κ(Sc ) = χ (Sc ) = 0. However, it is immediate that / Ba X . Furthermore, for any α-dense curve in Sc , say γ : I → X , Sc ∈ B X but Sc ∈ necessarily one has one and only one of the following possibilities: (a) γ (I ) ⊂ {(x, sin(1/x)) : x ∈ [−1, 0)}. (b) γ (I ) ⊂ {(x, sin(1/x)) : x ∈ (0, 1]}. (c) γ (I ) ⊂ {(0, y) : y ∈ [−1, 1]}. Therefore, in any of the three cases, we can find a point of Sc such that the distance from it to any point of γ (I ) is greater than 1. Hence for

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0 < α < 1 there is no α-dense curve in Sc . Consequently (Sc ) ≥ 1. This implies that  does not satisfy the property (i) of definition of MNC, so  is not a MNC. Moreover, taking into account that χ (Sc ) = 0, the inequality (Sc ) ≤ 2χ (Sc ) is not possible. Since κ(Sc ) = 0 and (Sc ) ≥ 1, it is obvious that the first inequality of (8) fails. In the next example we prove that the second inequality in (7) can be turned into an equality. Likewise we also demonstrate that  is not a MNC by pointing out that  does not satisfy the semi-additivity property of MNCs (see the definition of MNC).

Example This example appears in the proof of [22, Theorem 2.5]. Consider X = L 1 (I ), the space of Lebesgue-measurable functions defined on I and valued in R of integrable modulus endowed with the usual norm  f  := I | f (x)| d x. Define the set  B :=

 f (x)d x = 1 .

 f ∈ L 1 [0, 1] : f ≥ 0 with I

Then (B) = 2. Hence, since B ⊂ B X , where B X denotes the closed unit ball of X , and noticing it was proved in [38] that (B X ) = 1, it follows that  is not monotonous. Consequently  does not satisfy the property (iii) of the definition of MNC, so  is not a MNC. The most important properties of  are stated in the following result [22, Proposition 2.6]: Proposition 2.3 Let (X, ·) be a Banach space, B X the class of all nonempty bounded subsets of X and Ba X the subclass of all arc-connected sets of B X . Then  satisfies: (1) (B) = 0 if and only if B is a precompact set of Ba X . (2) (B) = (B) for all B ∈ B X , where B denotes the closure of B. (3) For B1 , B2 ∈ Ba X with B1 ∩ B2 = ∅, one has   (B1 ∪ B2 ) ≤ max (B1 ), (B2 ) . (4) (λB) = |λ| (B) for all λ ∈ R and for all B ∈ B X . (5) (x + B) = (B) for all x ∈ X and for all B ∈ B X . (6) (Conv(B)) ≤ (B), for all B ∈ B X where Conv(B) stands for the convex hull of B. As we have just seen  is not a MNC, but the above result points out that  shares with MNCs many properties. Now the question is if we can obtain by a natural way

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a MNC from . The answer is yes and it can be reached by defining the concept of degree of convex nondensifiability (DCND for short), denoted by c . Definition 2.5 The DCND of B ∈ B X is defined as   c (B) := sup (Conv(A) : A ⊂ B, A = ∅ ,

(9)

where Conv(A) denotes the convex hull of A. Firstly it is needed to prove that the DCND is well defined. Indeed, let B be a nonempty and bounded set of (X, ·). Then B is bounded and Conv(B) so is [31, Lemma B, p. 60]. Therefore, if we denote as B X the closed unit ball of X , there exists M > 0 such that Conv(B) ⊂ M B X . Let A = ∅ be such that A ⊂ B and γ : I → X an arbitrary continuous mapping with γ (I ) ⊂ Conv(A). Then, given x ∈ Conv(A), since Conv A ⊂ Conv(B), we have x − γ (t) ≤ x + γ (t) ≤ 2M, for arbitrary t ∈ I . This means that any curve γ in Conv(A) is 2M-dense in it. Hence, noticing (4), we have (Conv(A)) ≤ 2M, for all A ⊂ B, A = ∅. Then, from (9), it follows c (B) ≤ 2M. Consequently c (B) is well defined. The next result proves that c is a MNC, see [22, Theorem 3.3]. Theorem 2.5 Let (X, ·) be a Banach space and B X the class of all nonempty and bounded subsets of X . Then the degree of convex nondensifiability c defined in (9) is a MNC. The next result, see [22, Theorem 3.7], relates c with all measures of noncompactness in such a way that dominates to all them (see (10)). Theorem 2.6 Let (X, ·) be a Banach space, B X its closed unit ball and B X the class of all nonempty and bounded subsets of X . Let μ be an arbitrary MNC. Then the degree of convex nondensifiability c defined in (9) dominates μ, i.e. μ(B) ≤ μ(B X )c (B), for all B ∈ B X

(10)

holds. Moreover, by taking μ = χ (the Hausdorff MNC), the above inequality is the best possible in infinite dimensional Banach spaces.

3 A Generalization of D-Lipschitzian Mappings Through the DND In this section (X,  · ) is a Banach algebra satisfying

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x y ≤ xy for all x, y ∈ X . In [23], by using the DND, we deal with the problem of the existence of solutions of the equation A(x)B(x) + C(x) = x, (11) for certain operators defined in a nonempty, closed and bounded subset of X . Before to expose the result that guarantees the existence of some solution of (11) we need to introduce, in the below paragraphs, some concepts and results. We denote by R+ the interval [0, +∞) and consider the set   D := h : R+ → R+ : h is continuous, nondecreasing and h(0) = 0 .

(12)

Now we introduce the concept of D−Lipschitzian mapping, due to Dhage [14], as follows. Definition 3.1 Let be a nonempty closed convex subset of (X,  · ). A mapping T : → X is said to be D−Lipschitzian if there exists a function h of the set D defined in (12) such that  T (x) − T (y) ≤ h x − y , for all x, y ∈ . The function h is called a D-function of T . If, in addition, h satisfies h(r ) < r for all r > 0, then T is called a D-nonlinear contraction with a contraction function h. It is easy to check that a Lipschitzian mapping is in particular D-Lipschitzian, with function h(r ) := Lr , L > 0 being the Lipschitz constant of the mapping. However, √ the reciprocal is not true in general. Indeed, T : R → R defined as√T (x) = |x| is not Lipschitzian, but it is D−Lipschitzian for the function h(r ) = r ∈ D. The first connection between the degree of nondensifiability  and the D−Lipschitzian mappings is the following result, see [23, Proposition 3.1] Proposition 3.1 Let be a nonempty closed convex subset of (X,  · ), T : → X a D−Lipschitzian mapping with a contraction function h such that T ( ) ∈ B X and  the D N D. Then  (T (B)) ≤ h (B) for all B ∈ B X , B ⊂

(13)

holds. Inequality (13) allows us to define the concept of (, D)-Lipschitzian mapping. Definition 3.2 Let be a nonempty closed convex of (X,  · ), D the set defined in (12) and  the D N D defined on the family B X . A mapping T : → X such that T ( ) ∈ B X is said to be (, D)-Lipschitzian with a D function h if it satisfies inequality (13). The function h is then called a (, D)-function of T . If, in addition,

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h satisfies h(r ) < r for all r > 0, then T is called a (, D)-nonlinear contraction with a contraction function h. Proposition 3.1 points out that a D-nonlinear contraction is in particular a (, D)nonlinear contraction. Therefore this last class contains the class of D-nonlinear contractions. Moreover, it contains strictly. Indeed, a compact mapping T : → X which is not a D-nonlinear contraction is, trivially, a (, D)-nonlinear contraction. Less trivial is the following example. Example Let c0 be the space of the null sequences and Uc0 its closed unit ball. Fixed 0 < c < 1/2, consider T : Uc0 −→ Uc0 given by  T (x) := 1 − x, cx1 , . . . , cxn , . . .), for all x := (xn )n≥1 ∈ Uc0 . Then, if 0 is the null vector of c0 and x ∈ c0 with x = 1, as T (x) − T (0) = 1 = x − 0, we infer that T is not a D-nonlinear contraction. Note that we can write T (x) = K (x) + S(x), with   K (x) := 1 − x, 0, . . . , 0, . . .), S(x) := 0, cx1 , . . . , cxn , . . .). Therefore, as K is compact and S is a D-nonlinear contraction with function h(r ) := cr , from Proposition 3.1 and Theorem 2.4 we have 

    T (B) ≤ 2 χ K (B) + χ S(B) ] = 2χ S(B) ≤ 2c(B), for each nonempty and convex set C ⊂ Uc0 . Therefore T is a (, D)-nonlinear contraction with function h(r ) := 2cr . Other example of a (, D)-nonlinear contraction which is not a D-nonlinear contraction can be found in [23, Example 3.1]. A key result to prove the existence of solutions of (11) is the following (see [23, Theorem 3.1]). As in previous sections, B X denotes the class of nonempty and bounded subsets of X . Theorem 3.1 Let ∈ B X be a closed and convex set, and T : −→ a (, D)nonlinear contraction with a contraction function h ∈ D. Then T has some fixed point. At this point, we recall the celebrated Krasnosel’ski˘ı fixed point result [33], which states that if ∈ B X is closed and convex, A, C : −→ X continuous and the conditions: (1) A(x) + C(x) ∈ , for each x ∈ . (2) A is a k-contraction, for some k ∈ [0, 1), i.e. A(x) − A(x) ≤ kx − y for all x, y ∈ .

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(3) C is compact. hold, then the equation A(x) + C(x) = x has solution. This result can be generalized, in our context as a consequence of Theorem 3.1, in the following way: Corollary 3.1 Let ∈ B X be a closed and convex set and A, C : −→ X continuous. Assume the conditions: (1) A(x) + C(x) ∈ , for each x ∈ . (2) A is (, D)-Lipschitzian with a D-function h A . (3) C is compact. In addition, suppose that 2h A ((S)) < (S), for each nonempty and convex set S ⊂ with (S) > 0. Then, the equation A(x) + C(x) = x has solution. On the other hand, the main result of [23] is the following, which states sufficient conditions to ensure the existence of solutions of the Eq. (11). Theorem 3.2 Let ∈ B X be a closed and convex set and A, B, C : −→ X continuous. Assume the conditions: (1) A(x)B(x) + C(x) ∈ , for each x ∈ . (2) A, C are (, D)-Lipschitzian with D-functions h A and h C , respectively. (3) B is compact. In addition, suppose that 

2 Bh A ((S)) + h C ((S)) < (S),

(14)

for each nonempty and convex set S ⊂ with (S) > 0. Then, the equation A(x)B(x) + C(x) = x has some solution x ∈ . It is important to stress that the conditions of the above theorem are more general than other existence results. Indeed, as we point out in [23]: (I) The mappings A and C are (, D)-Lipschitzian instead D-Lipschitzian as required in many works cited in [23, Sect. 1]. Such works are based on DLipschitzian conditions (which, in view of Proposition 3.1 are particular cases of (, D)-Lipschitzian mappings), or in the existence of the inverse of some mapping A, B, C. (II) The class of (, D)-nonlinear contractions is larger than others classes of mappings used in some generalizations of Darbo fixed point theorem (see, for instance, [2, Theorem 3]). As consequence of the above theorem, we have the following:

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Corollary 3.2 Let ∈ B X be closed and convex and A, B, C : −→ X continuous. Assume the conditions: (1) A(x)B(x) + C(x) ∈ for each x ∈ . (2) A is (, D)-Lipschitzian with a D-function h A . (3) B and C are compact. In addition, assume that 2Bh A ((S)) < (S) result (compare the inequality that is assumed in it with the inequality (14)) for each nonempty and convex set S ⊂ with (S) > 0. Then the equation A(x)B(x) + C(x) = x has some solution x ∈ . The DND has been successful used to prove the existence of solutions, under suitable conditions, of certain integral equations. Such integral equations posed in Banach spaces. Indeed, in [20] we prove the existence of solutions of infinite systems of integral equations, by posing such system as an integral equation into a suitable sequence space. In [21], we solve the problem c

D p x(t) = f (t, x(t)), t ∈ [0, T ] , x(0) = x0

p

where c D p :=c D0+ , 0 < p < 1, denotes the Caputo fractional derivative of order 0 < p < 1, with the lower limit zero, x0 ∈ X and f : [0, T ] × X −→ X are given, X being a Banach space. To prove the existence of solutions of the above problem, we pose it as the integral equation 1 x(t) = x0 + ( p)



t

(t − s) p−1 f (s, x(s))ds for all t ∈ [0, T ],

0

where (·) stands for the usual Gamma function. Under suitable conditions on the function f , the existence of solutions of the above integral equation is derived from some fixed point result based on the DND. Other types of integral equations which can be solved by using some fixed point result based on the DND are shown in Sect. 4. Corollary 3.2 allows us to prove, under suitable conditions, the existence of solutions for quadratic integral equations of the form



x(t) = q(t) + f (t, x(t)) + 0

p1 (t)

K 1 (t, s, x(s))ds



p2 (t)

K 2 (t, s, x(s))ds,

0

for each t ∈ [0, T ], T > 0 fixed, where q : [0, T ] −→ R, f : [0, T ] × R −→ R and pi : [0, T ] −→ [0, T ] , K i : [0, T ]2 × R −→ R for i = 1, 2 are known. A particular case of the above integral quadratic equations is the so called Ambartsumian-Chandrasekhar integral equation

The Degree of Non-densifiability of Bounded Sets of a Banach …



1

x(t) = q(t) + x(t) 0

149

t x(s)ds for all t ∈ [0.1], t +s

which is used in the theory of radioactive transfer in semi-infinite atmospheres, can be proved by using Corollary 3.2.

4 The DND on Projective Limits of Generalized Scales of Banach Spaces and Applications In this section we introduce the notion of generalized scale of Banach spaces (GSBS for short) which is more general than the scale of Banach spaces, written SBS for short. On the SBS concept and applications see for instance [16, 46, 47]. The spaces of a GSBS define a projective limit space denoted by X . In that space will be defined a type of operators based on the DND. Such operators will be called DND-condensing operators. It is shown that these operators have a fixed point in X . In this manner we obtain a method for the Theory of Fixed Point. Some applications of DND-condensing operators have been proposed. Firstly let us agree all Banach spaces considered along this section are non-trivial, i.e. they do not reduce to {0}. The notion of GSBS is defined as follows: Definition 4.1 A family of Banach spaces (X s ,  · s )00



t∈Tε (x)

 ∂ε f t (x) + Ndom f (x) ,

as X ∈ F(x), and Ndom f (x) ⊂ N L∩dom f (x), for all L ∈ F(x). Some characterizations of Ndom f (x) are available e.g. in [25, Lemma 5]:

  x ∗ ∈ Ndom f (x) ⇔ (x ∗ , x ∗ , x) ∈ co ∪t∈T gph f t∗ ∞ , where [·]∞ is the recession cone. In particular, if f (x) := sup{at∗ , x − bt : t ∈ T }, at∗ ∈ X ∗ , bt ∈ R, we get 

 x ∗ ∈ Ndom f (x) ⇔ (x ∗ , x ∗ , x) ∈ co (θ, 0) ∪ {(at∗ , bt ), t ∈ T } ∞ , and, for every x ∈ X, ∂ f (x) =

 L∈F (x),ε>0

cl co{at∗ : t ∈ Tε (x)} + B L ,

where 

 x ∗ ∈ B L ⇔ (x ∗ , x ∗ , x) ∈ co (L ⊥ × {0}) ∪ {(at∗ , βt ), t ∈ T } ∞ . (See [39] for extensions of this formula.) Condition (19) is extended in [41, Theorem 4] to a family of general functions, non necessarily convex, f t : X → R, t ∈ T, such that dom f = ∅ and

A New Tour on the Subdifferential of the Supremum Function

∗∗  f ∗∗ ≡ sup f t = sup f t∗∗ . t∈T

177

(CC)

t∈T

Then, at every x ∈ X , we have ⎛



∂ f (x) =



co ⎝

ε>0, z∈dom f

⎞ ∂ε f t (x) + {z − x}− ⎠ .

t∈Tε (x)

A version of main formula (17) is given next, in (20), when X is a Banach space and the functions f t : X → R, t ∈ T, are convex and lsc. Like in (15), ∂ f is expressed now by using the (exact) subdifferentials of the data functions f t , t ∈ Tε (x), but at nearby points. More precisely, for every x ∈ X, ∂ f (x) =

where

 ε>0 L∈F (x)

 co





t∈Tε (x),y∈Bt (x,ε)

∂ f t (y) ∩ Sε (y − x) + N L∩dom f (x) ,

(20)

Sε (y − x) := {y ∗ ∈ X ∗ : y ∗ , y − x ≤ ε},

(21)

Bt (x, ε) := {y ∈ X : y − x ≤ ε and | f t (y) − f t (x)| ≤ ε}.

(22)

and

The following example draws aside the possibility of extending (20) to non-Banach spaces. Given a lcs X , let us consider a function g ∈ Γ0 (X ) having an empty subdifferential everywhere (such a function exists as it is seen in [4, 47]). Obviously, we may suppose that θ ∈ dom g and g(θ) = 0. If we introduce the functions f t ∈ Γ0 (X ), t ∈ T := ]0, +∞[ , defined by f t (x) := tg(x), it turns out that f := supt∈T f t = I[g≤0] . Since ∂ f t ≡ t∂g ≡ ∅, for all t ∈ T, the right-hand set in (20) is empty, whereas ∂ f (θ) = N[g≤0] (θ) = ∅. Formula (23) below gives an alternative to (17), without assuming condition (19). The price to pay is to use the augmented functions f t + I L∩dom f , t ∈ T , instead of the original f t ’s in the main formula: Given the convex functions f t : X → R, t ∈ T, for every x ∈ X we have ∂ f (x) =

 ε>0 L∈F (x)

 co



t∈Tε (x)

 ∂ε ( f t + I L∩dom f )(x) .

(23)

We may have the impression that (23) is simpler than (17) as N L∩dom f (x) disappeared but when we try to decouple the subdifferential of the sum (applying for instance (7) if the functions f t belong to Γ0 (X )), it leads to results of similar complexity. At this point, let us anticipate that under the so-called compactness/continuity assumptions, standing for the compactness of the index set T and the upper semi-

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A. Hantoute and M. A. López-Cerdá

continuity of the mappings t ∈ T → f t (z), z ∈ X, we get ∂ f (x) =





co

L∈F (x)

 t∈T (x)

 ∂( f t + I L∩dom f )(x) ;

in other words, under these assumptions, the closure and the intersection over ε in (23) are removed.

3.2 The Role of Continuity Assumptions When the supremum function f exhibits a certain degree of continuity, the formulas for ∂ f (x) are greatly simplified; in fact, the finite-dimensional sections of dom f used in the previous section are removed. More precisely, in [25] (see, also, [30, 41]), it is proved that, given a point x ∈ dom f such that either ri(cone(dom f − x)) = ∅ or cone(dom f − x) is closed, then, provided that (19) holds, we have that ∂ f (x) =



 co

ε>0





t∈Tε (x)

∂ε f t (x) + Ndom f (x) .

If the set int(cone(dom f − x)) is non-empty, or if f is finite and continuous somewhere, then    co (24) ∂ f (x) = t∈Tε (x) ∂ε f t (x) + Ndom f (x). ε>0

If f is finite and continuous at the reference point x, then we get (16), and ∂ f (x) =

 ε>0, p∈P

⎧ ⎨ co

⎩

 t∈Tε (x),y∈B p (x,ε)

⎫ ⎬

∂ f t (y) , ⎭

(25)

where P is a (saturated) family of continuous seminorms that generates the topology of X and B p (x, ε) := {y ∈ X : p(y − x) ≤ ε}.

3.3 The Role of Compactness-Continuity Assumptions In a variety of applications, including the functional approach, semi-infinite programming, etc., the following assumptions are standard: T is compact and the mapping t → f t (z) is upper semicontinuous (usc, in brief) for all z ∈ dom f. These assumptions ensure that (see [14]) T (x) = ∅, dom f = ∩t∈T dom f t , and, for every x ∈ dom f,

A New Tour on the Subdifferential of the Supremum Function

179

R+ (dom f − x) = ∩t∈T R+ (dom f t − x). In [13, 14], we considered a family of convex functions f t , t ∈ T, satisfying these compactness/continuity assumptions, and proved that, for every x ∈ dom f, ∂ f (x) =



co

L∈F (x)

⎧ ⎨  ⎩

t∈T (x)

⎫ ⎬

∂( f t + I L∩dom f )(x) . ⎭

(26)

Oberve that this expression holds without assuming condition (19). The compactness of T can be relaxed by taking, instead of the whole set T, the subset Tε (x) for sufficiently small ε > 0. If the functions are lsc we actually get ∂ f (x) =



co

ε>0 L∈F (x)

⎧ ⎨  ⎩

t∈T (x)

⎫ ⎬

∂ε f t (x) + N L∩dom f (x) . ⎭

If the supremum function f := supt∈T f t is continuous at x, then we deduce (13). Moreover, if the functions f t ’s are finite and continuous on an open set U ⊂ X , (13) holds on U. Similar results can be found in [32], [51], and [54]. We complete this subsection with some alternative conditions which are of geometrical and topological nature and depend on the way in which the functions involved are related, namely, whether their effective domains overlap sufficiently. The term qualification applies here because the conditions we give are of the same type as those used in the section below to derive optimality conditions for convex optimization problems. The following condition is a finite-dimensional qualification. Instead of assuming the continuity of the supremum function f , it relies on the effective domains of the f t ’s. More precisely, given the convex functions f t : Rn → R∞ , t ∈ T, and f := supt∈T f t , take x ∈ dom f , and assume that the standard compactness/continuity assumptions hold. If additionally we assume that ri(dom f t ) ∩ dom f = ∅, for all t ∈ T (x), 

then ∂ f (x) = co

 t∈T (x)

(27)

 ∂ f t (x) + Ndom f (x).

(28)

It is easily verified that (28) remains valid if, instead of condition (27), we require that t∈T ri(dom f t ) = ∅. If we are pursuing the goal of making disappear N L∩dom f (x), the following result is useful (see [15, Theorem 5]). If { f t , t ∈ T } ⊂ Γ0 (X ) is such that, at x ∈ dom f , we have inf t∈T f t (x) > −∞, under the compactness/continuity assumptions, the following formula holds true

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!  "  ∂ε f t (x) , for every ε > 0. Ndom f (x) = co

(29)

∞

t∈T

Formula (29) reminds us formula (3), Ndom f (x) = (∂ε f (x))∞ , but (29) is more operative as the formulas giving an expression of ∂ε f (x) are quite involved. For instance, in [17, Theorem 4.1.3], we proved the following result representing ∂ε f (x) through convex combinations of finite subfamilies: If { f t , t ∈ T } ⊂ Γ0 (X ), for all x ∈ dom f and ε ≥ 0, we have that, under the convention 0 f t = Idom ft , ∂ε f (x) =

 δ>ε

 cl





λ∈Δk ,(ti )1≤i≤k ⊂T,k≥1

∂δ+1≤i≤k λi fti (x)− f (x)

#

 λi f ti

 (x) . (30)

1≤i≤k

(See, also, [44, Proposition 3.1].) In particular, the ε−subdifferential of the supremum of affine functions, f := supt∈T (at , · − bt ), is easily obtained by applying (6): For every x ∈ X and ε ≥ 0,     ∂ε f (x) = x ∗ ∈ X ∗ : (x ∗ , α∗ ) ∈ co(C), x ∗ , x − α∗ ≥ f (x) − ε ,

(31)

where (∅ =)C := {(at , bt ), t ∈ T } ⊂ X ∗ × R. Moreover, (29) allowed us to deduce in the compact framework (see [15, Theorem 10])        co ∂ε f t (x) + {0, ε}∂ε f t (x) . ∂ f (x) = ε>0

t∈T (x)

Hence, if T (x) = T, ∂ f (x) =

t∈T \T (x)

 ε>0

 co



 ∂ε f t (x) .

(32)

t∈T

Formula (32) is an extension to infinitely many functions of the Brøndsted formula (14).

3.4 Compactifying When the compactness/continuity assumptions fail, we can apply compactification strategies consisting of compactifying T and enlarging the set of functions by taking upper limits of the mappings t → f t (z), z ∈ dom f. In a first step, we identify T as a subset of the Hausdorff compact product space S := [0, 1]C(T,[0,1]) ≡ {γ : C(T, [0, 1]) → [0, 1]}, where C(T, [0, 1]) is the space of continuous functions defined on T and valued in [0, 1]. Hence, the convergence of the net (γi )i ⊂ S to γ ∈ S, written γi → γ,

A New Tour on the Subdifferential of the Supremum Function

181

is the pointwise convergence, in other words, if and only if γi (ϕ) → γ(ϕ) for all ϕ ∈ C(T, [0, 1]). The identification of T as a subset of S is made by means of the mapping w : T → S, defined as (33) w(t) ≡ γt , where γt , t ∈ T, is the evaluation function, i.e., γt (ϕ) := ϕ(t), ϕ ∈ C(T, [0, 1]). ˇ Then, the Stone-Cech compact extension of T is βT := cl(w(T )), i.e. the closure of w(T ) with respect to the product topology. It is well-known that βT is a Haussdorff compact subset of S and that the mapping w is continuous. Moreover, if T is Tychonoff (completely regular and Hausdorff) then w is an homeomorphism between T and βT , that is, γti → γt if and only if ti → t in T, for every t ∈ T and net (ti )i ⊂ T. The second step of the compactification process consists of regularizing the original functions f t , t ∈ T. To this aim, given γ ∈ βT, we define the function f γ : X → R as (34) f γ (x) := lim sup f t (x). γt →γ,t∈T

  Obviously, the family f γ , γ ∈ βT includes all the elements of the form f γt , t ∈ T, defined as f γt (x) = lim supγs →γt , s∈T f s (x). If T is Tychonoff, f γt (x) = lim sups→t,s∈T f s (x), and these functions may not belong to the original family f t , t ∈ T. In [14] we proved that the functions f γ , γ ∈ βT, are convex and satisfy f = supt∈T f t = maxγ∈βT f γ . Moreover, the mappings γ → f γ (z), z ∈ dom f , are usc. If we define, for x ∈ f −1 (R) and ε ≥ 0, the extended ε-active index set of f at x   $ε (x) := γ ∈ βT : f γ (x) ≥ f (x) − ε , T

(35)

$(x) := T $0 (x), in that paper we also as well as the extended active index set T $(x) ⊃ w(T (x)) and T $(x) = $ε (x) is non-empty and compact, that T proved that T  cl (x))) . Finally, if the compactness/continuity assumptions are fulfilled (w(T ε ε>0  $(x) = w(T (x)) = ε>0 w(Tε (x)). one has T Having at hand all the elements that we have just introduced, we can apply the previous results that were established under the conditions of compactness/continuity. In [14] we characterized ∂ f (x) in terms of (exact) subdifferentials involving the $(x). Actually, given the convex functions f t : X → R, t ∈ T, and f γ ’s, γ ∈ T f := supt∈T f t , where T is a topological space, for every x ∈ dom f we have that ∂ f (x) =

 L∈F (x)

co

⎧ ⎨  ⎩

$(x) γ∈T

⎫ ⎬

∂( f γ + I L∩dom f )(x) . ⎭

(36)

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In fact, the proof of (36) is a direct consequence of the following facts: the functions f γ , γ ∈ βT, are convex, their supremum is f , the set βT is compact in S and the mappings γ → f γ (z), z ∈ dom f , are usc. Therefore, (26) immediately applies. The following result provides us with an explicit reformulation of the last result, in which the elements from the set [0, 1]C(T,[0,1]) are now replaced by nets in T : ∂ f (x) =



co

L∈F (x)

⎧ ⎨  ⎩

(ti )i ∈T (x)

⎫

⎬ ∂ lim f ti + I L∩dom f (x) , i ⎭



(37)

where each limit limi ( f ti + I L∩dom f ) exists in R ∪ {+∞} in a convex neighborhood of x, and (38) T (x) := {(ti )i ⊂ T : lim f ti (x) = f (x)}. i

With the purpose of illustrating the application of the compactification process that we have just described, we present the following example. Consider the family of convex functions g2n+1 , h 2n : X → R, n ∈ N, defined on R as % nx ,0 , g2n+1 (x) := max n+1 

% −nx h 2n (x) := max ,0 . n+1 

We gather these functions in the family { f n , n ∈ N} such that f 2n+1 := g2n+1 and f 2n := h 2n , and take the supremum function f := supn∈N f n = sup {g2n+1 , h 2n } . n∈N Obviously, f (x) = |x| , but if we apply the formula ∂ f (x) = n∈T (x) ∂ f n (x) we reach a false conclusion (obviously, N is not compact):  ∂ f (x) =

]−1, 1[ , if x = 0, ∅, if x = 0.

If we apply formula (37) at x = 0, we get ∂ f (0) = co

⎧ ⎨  ⎩

(ti )i ∈T (0)

⎫ ⎬

∂(lim f ti )(0) , i ⎭

(39)

where T (0) = {(ti )i ⊂ T : limi f ti (0) = 0}, and each limit limi f ti exists in R ∪ {+∞} in a convex neighborhood of 0. In this case, only four cases arise: a) limi f ti = g2n 0 +1 , for n 0 ∈ N ( f ti = g2n 0 +1 , eventually), b) limi f ti = h 2n 0 , for n 0 ∈ N ( f ti = h 2n 0 , eventually), c) limi f ti = max{z, 0} =: gγ¯ (z) (ti is eventually odd), d) limi f ti = max{−z, 0} =: h γ¯ (z) (ti is eventually even). So, (39) yields

A New Tour on the Subdifferential of the Supremum Function

∂ f (0) = co

⎧ ⎨ 

183

⎫ ⎬

∂(lim f ti )(0) i ⎩ ⎭ (ti )i ∈T (0)      = co (∂g2n+1 (0) ∪ ∂h 2n (0)) ∂gγ¯ (0) ∂h γ¯ (0) n≥1

 = co

! 0,



n≥1

 " ! " n n ∪ − , 0 ∪ {1} ∪ {−1} n+1 n+1

= [−1, 1]. ˇ The results above based on the Stone-Cech compactification also lead us to formulas for ∂ f (x) which involve the one-point compact extension of the index set T . This extension is performed through the following procedure. Given a topology τ on T, and an element ω ∈ / T, we consider the compact topological space T ∪ {ω}, with topology given by τ ∪ {{ω} ∪ (T \ C) : C ∈ C(T )},

(40)

where C(T ) := {C ⊂ T : C is compact and closed}. (Remember that T and T ∪ {ω} are not required to be Hausdorff, and that the onepoint compactification is particularly adequate whenever T is locally compact.) In the following result, f˜ω : X → R denotes the convex function defined by f˜ω := lim sup f s ,

(41)

s→ω,s∈T

that is, denoting by V(ω) the family of neighborhoods of ω,     f˜ω = inf sup f s = inf sup f s . V ∈V(ω)

C∈C(T )

s∈V

s∈T \C

Observe that f˜ω ≤ f , and in the example above f˜ω (x) = |x| . We also introduce the usc regularizations f˜t : X → R, t ∈ T, defined by f˜t := lim sups→t f s and consider T˜ (x) := {t ∈ T : f˜t (x) = f (x)}. Then, for every x ∈ dom f we have that ∂ f (x) = where

 L∈F (x)

 co

 s∈T ω (x)

 ∂( f˜s + I L∩dom f )(x) ,

(42)

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ω

T (x) :=

T˜ (x), if f˜ω (x) < f (x), T˜ (x) ∪ {ω}, if f˜ω (x) = f (x).

If T = N one gets ∂ f (x) =







co

L∈F (x)

s∈T ω (x)

 ∂( f s + I L∩dom f )(x) .

4 Some Applications in Convex Analysis In this section we present some applications of the results in the previous section, in the framework of convex analysis. The first one deals with the subdifferential of the sum, both under symmetric and asymmetric assumptions. The second subsection analyzes the possibility of convexifying a general (unconstrained) optimization problem and studies the relationship between the optimal set of the convexified problem and the approximate or exact optimal solutions of the original one.

4.1 Subdifferential of the Sum In [25, Theorem 13], and based on our main formula (17), we proved the HiriartUrruty and Phelps formula ([29]) ∂(g + f ◦ A)(x) =

 ε>0



cl ∂ε g(x) + A∗ ∂ε f (Ax) ,

where f : Y → R and g : X → R are convex functions, X and Y are separated lcs spaces, A : X → Y a continuous linear mapping with continuous adjoint A∗ , but relaxing the assumption f ∈ Γ0 (Y ), g ∈ Γ0 (X ) to the slightly weaker closednesstype assumption cl(g + f ◦ A) = (cl g) + (cl f ) ◦ A. (43) (In fact, (43) constitutes a counterpart of the closedness condition (19) for the sum operation.) Let us give here a sketch of the proof, in order to illustrate the application of (17). While the inclusion “⊃” trivially holds, we proceed by proving “⊂”. If we define ϕ := g + f ◦ A and ψ := (cl g) + (cl f ) ◦ A, when ∂ϕ(z) = ∅, we have

A New Tour on the Subdifferential of the Supremum Function

185

(43)

(cl g) (z) + (cl f ) (Az) = (cl ϕ) (z) = ϕ(z) = g(z) + f (Az) ∈ R, ∂ϕ(z) = ∂ (cl ϕ) (z) = ∂ ((cl g) + (cl f ) ◦ A) (z) = ∂ψ(z), and cl f ∈ Γ0 (Y ) and cl g ∈ Γ0 (X ). By linearizing cl f , we write for x ∈ X ψ(x) = (cl g) (x) + sup{y ∗ , Ax − f ∗ (y ∗ ) | y ∗ ∈ dom f ∗ } = sup{(cl g) (x) + A∗ y ∗ , x − f ∗ (y ∗ ) | y ∗ ∈ dom f ∗ }, and we are in situation to apply (17) giving rise to the aimed inclusion. If we additionally assume that f is continuous at some point in A(dom g), then, for every x ∈ dom g ∩ A−1 (dom f ) and ε ≥ 0, the set ∂ε g(x) + A∗ ∂ε f (Ax) is proved to be closed, and this yields the classical chain rule by Moreau and Rockafellar for the sum and composition with a continuous linear mapping (see, e.g., [43]): ∂(g + f ◦ A)(x) = ∂g(x) + A∗ ∂ f (Ax), for every x ∈ X. In [9] some asymmetric subdifferential sum rules were established. They are given in terms of the exact subdifferential of one function (the most qualified one), and the approximate subdifferential of the other one, and require that the domains of f and g overlap sufficiently, or that the epigraphs enjoy certain closedness-type properties. More precisely, in [9, Theorem 12] we proved that for f, g ∈ Γ0 (X ) and x ∈ dom f ∩ dom g, if at least one of the following conditions, involving the effective domains and the epigraphs, holds: (i) R+ (epi g − (x, g(x))) is closed, (ii) dom f ∩ ri(dom g) = ∅ and g|aff(dom g) is continuous on ri(dom g), then ∂( f + g)(x) =

 ε>0

cl(∂ε f (x) + ∂g(x)).

(44)

Observe that if the set epi g is polyhedral, then condition (i) holds. We even established in such a paper the quasi-exact subdifferential rule ∂( f + g)(x) = cl (∂ f (x) + ∂g(x)) , under any one of the following assumptions: (iii) R+ (epi f − (x, f (x))) is closed, dom f ∩ ri(dom g) = ∅, and g|aff(dom g) is continuous on ri(dom g), (iv) R+ (epi f − (x, f (x))) and R+ (epi g − (x, g(x))) are closed, (v) ri(dom f ) ∩ ri(dom g) = ∅ and f |aff(dom f ) and g|aff(dom g) are continuous on ri(dom f ) and ri(dom g), respectively. Moreover, under (v), if ∂ f (x) or ∂g(x) is locally compact, then ∂( f + g)(x) = ∂ f (x) + ∂g(x). The last symmetric results can be found in [9, Theorem 15], and can be seen as an infinite-dimensional extension of Theorem 23.8 in [48]. In particular, condition (iii) can be regarded as a counterpart of the Attouch-Brézis condition [1] for general locally convex spaces. Compare, also, with [3, Corollary 4.3] where the so-called quasi-relative interior notion is involved.

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In [10, Theorem 4] the subdifferential of the sum f + g, when f := supt∈T f t , and g, f t : X → R∞ , t ∈ T (= ∅), are all proper convex functions, under the assumption cl ( f + g) (x) = sup(cl f t )(x) + g(x) , for all x ∈ dom f ∩ dom g, t∈T

is characterized by means of the following extension of (17): ∂( f + g)(x) =

 L∈F (x),ε>0

 co

 t∈Tε (x)

 ∂ε f t (x) + ∂(g + I L∩dom f )(x) .

4.2 Convexifying Given a proper function f : X → R∞ , we consider the general unconstrained optimization problem (P) Min f (x), s.t. x ∈ X, together with its associated convex relaxation (Pr ) Min (co f )(x), s.t. x ∈ X, By v(P) and v(Pr ) we denote the optimal values of (P) and (Pr ), respectively. Such a convexification process does not alter the optimal value of the original problem (P); i.e., the values v(P) and v(Pr ) coincide, and every solution of (P) is a solution of (Pr ). Our purpose here is then to express the optimal set of (Pr ) in terms of the approximate and/or exact optimal solutions of (P). To this aim, we consider the set of ε-optimal solutions of (P), ε ≥ 0, defined by ε- argmin f := (∂ε f )−1 (θ) = {x ∈ X : f (x) ≤ v(P) + ε}; in particular, we put argmin f := 0-argmin f. Due to the properness assumption on f, we have that ε-argmin f = ∅ whenever v(P) = −∞, while ε-argmin f = ∅ when ε > 0 and v(P) > −∞. These relaxation arguments are rather useful in practice, namely, in calculus of variations, in mathematical programming problems, as well as in many other theoretical and numerical purposes (see, e.g., [21, 28]). This fundamental topic has been considered by many researchers in recent years. The approach in [2] is based on the subdifferential analysis of the closed convex hull. The following results, given in [41], express the optimal set of (Pr ) by means of the approximate optimal solutions of (P), and are based on the subdifferential of the Fenchel conjugate, which itself relies on the subdifferential of a supremum function. For every function f : X → R∞ with a proper conjugate, we have

A New Tour on the Subdifferential of the Supremum Function

argmin(co f ) =

 ε>0,L∈F X ∗



co ε- argmin f + N L∩dom f ∗ (θ) ,

187

(45)

where F X ∗ is the family of finite-dimensional linear subspaces of X ∗ . Moreover, the following statements hold true: (i) If ri(cone(dom f ∗ )) = ∅ or cone(dom f ∗ ) is closed, then argmin(co f ) =

 ε>0



co ε- argmin f + Ndom f ∗ (θ) ,

(46)

and, when additionally cone(dom f ∗ ) = X ∗ , argmin(co f ) =

 ε>0

co (ε- argmin f ) .

(ii) If int(cone(dom f ∗ )) = ∅, then argmin(co f ) = Ndom f ∗ (θ) +

 ε>0

co (ε- argmin f ) .

These results apply to derive the following expressions of ∂ f ∗ (x ∗ ) in [41, Theorem 3]. Given a function f : X → R such that dom f ∗ = ∅, one has for all x ∗ ∈ X ∗ , ∂ f ∗ (x ∗ ) =

 ε>0, u ∗ ∈dom f ∗

co (∂ε f )−1 (x ∗ ) + {u ∗ − x ∗ }− .

Moreover, if ri(cone((dom f ∗ ) − x ∗ )) = ∅ or cone ((dom f ∗ ) − x ∗ ) is w ∗ −closed, then 

co (∂ε f )−1 (x ∗ ) + Ndom f ∗ (x ∗ ) . ∂ f ∗ (x ∗ ) = ε>0

Finally, if {Ci , i ∈ I } is a family of convex sets of X ∗ satisfying dom f ∗ ⊆ ∪i∈I Ci , and

ri cone(Ci ∩ dom f ∗ ) = ∅, for all i ∈ I, one has argmin(co f ) =



co (ε − argmin f ) + (Ci ∩ dom f ∗ )− .

ε>0, i∈I

If we take {Ci , i ∈ I } = {{x ∗ }, x ∗ ∈ dom f ∗ }, we get (46).

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5 Optimality Conditions in Convex Optimization This section deals with the convex optimization problem (P) Min g(x) s.t. f t (x) ≤ 0, t ∈ T, x ∈ C, where T is an arbitrary index set, C is a non-empty closed convex subset of a (separated) lcs X , and g, f t : X → R ∪ {+∞}, t ∈ T, all belong to Γ0 (X ). We assume that the constraint system S := { f t (x) ≤ 0, t ∈ T ; x ∈ C},

(47)

is consistent; i.e., it has a non-empty set of feasible solutions, which is represented by F. An important particular case is that in which the explicit constraints are affine and   continuous and there is no constraint set C (equivalently, C = X ); i.e., S = { at∗ , x ≤ bt , t ∈ T }, where at∗ ∈ X ∗ , t ∈ T . When T is infinite, the objective function g is linear, and X = Rn , this is the so-called linear semi-infinite optimization problem (LSIP, in brief). Our goal in this section is twofold, more precisely, we present, in the current framework of infinite convex systems, some constraint qualifications as the FarkasMinkowski property and the local Farkas-Minkowski property, and derive optimality conditions for (P) by appealing to the subdifferential of the supremum function of infinitely convex functions. We call characteristic cone of the constraint system S = { f t (x) ≤ 0, t ∈ T ; x ∈ C} to the convex cone  K := cone



 epi f t∗ ∪ epi σC

 = cone

t∈T



 epi f t∗ + epi σC .

(48)

t∈T

The cone K plays a crucial role in our theory. For the linear system considered above, epi f t∗ = (at∗ , bt ) + R+ (θ, 1), and epi σC = epi σ X = R+ (θ, 1); hence,   K = cone (at∗ , bt ), t ∈ T ; (θ, 1) .

(49)

It is straightforward to see that, if F = {x ∈ C : f t (x) ≤ 0, t ∈ T } = ∅, then epi σF = cl K = cl cone

& t∈T

' epi f t∗ ∪ epi σC .

In fact, if h := sup{ f t , t ∈ T ; IC }, we have x ∈ F ⇔ h(x) ≤ 0 ⇔ h(x) = 0. Then, by (9),

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  h ∗ = {sup { f t , t ∈ T ; IC }}∗ = co inf f t∗ , t ∈ T ; σC , and

(∗)

epi σF = cl(cone epi h ∗ ) = cl K,

where (∗) follows from Lemma 3.1(b) in [33], a result which is also valid in a lcs. The following result constitutes a sort of a generalized Farkas lemma. Given ϕ, ψ ∈ Γ0 (X ), then ϕ(x) ≤ ψ(x) for all x ∈ F, assumed non-empty, if and only if

epi ϕ∗ ⊂ cl epi ψ ∗ + K .

(50)

The proof is straightforward. Actually, ϕ(x) ≤ ψ(x) ∀x ∈ F ⇔ ϕ ≤ ψ + IF ⇔ (ψ + IF )∗ ≤ ϕ∗ ⇔ epi ϕ∗ ⊂ epi (ψ + IF )∗ , but applying (11), the previous lemma, and cl(A + B) = cl(A + cl B), we write epi (ψ + IF )∗ = cl(epi ψ ∗ + epi σF ) = cl(epi ψ ∗ + cl K) = cl(epi ψ ∗ + K). By applying the generalized Farkas lemma to ϕ = a ∗ , · − α (with a ∗ ∈ X ∗ , α ∈ R) and ψ ≡ 0, we see that the inequality a ∗ , x ≤ α holds for all x ∈ F, assumed non-empty (i.e., a ∗ , x ≤ α is a continuous linear consequence of S), if and only if ∗ a , α ∈ cl K. The following property is crucial in getting KKT-type optimality conditions for problem (P). We say that the consistent system S = { f t (x) ≤ 0, t ∈ T ; x ∈ C}) is Farkas-Minkowski (FM, in brief) if K is w∗ -closed. This property was introduced by Charnes, Cooper and Kortanek in [8] as a general assumption for the duality theory in LSIP (see also [23]). The FM property for convex systems was first studied in [34], with X being Banach and all the functions finite valued, under the name of closed cone constraint qualification. The FM property is strictly weaker than several known interior-type regularity conditions. If S is FM, then it turns out that every continuous linear consequence a ∗ , x ≤ α of S is also consequence of a finite subsystem. The converse statement holds if S is linear. In [20] the following KKT-type optimality conditions for the problem (P), having a non-empty set of feasible solutions F, are derived. If we assume that the constraint system is FM and that g is continuous at some point of F, then x ∈ F ∩ dom g is a ) global minimum of (P) if and only if there exists λ ∈ R(T + such that ∂ f t (x)  = ∅, ∀t ∈ supp λ, and the KKT conditions

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θ ∈ ∂g(x) +

(

λt ∂ f t (x) + NC (x) and λt f t (x) = 0, ∀t ∈ T,

(51)

t∈T ) hold. Here R(T + is the convex cone of all the functions λ : T → R+ which vanishes at every point of T except at finitely many, and supp λ := {t ∈ T : λt > 0}. Let us give a sketch of the proof. The point x will be a minimizer of (P) if and only if (∗ )

θ ∈ ∂(g + IF )(x) = ∂g(x) + ∂IF (x) = ∂g(x) + NF (x);

(52)

i.e., if and only if there exists x ∗ ∈ ∂g(x) such that x ∗ , x ≥ x ∗ , x is consequence of S. The equality (∗ ) comes from the continuity of g at some point of F ≡ dom IF . Now, if x is a minimizer of (P), since S is FM we have  ∗

∗

−(x , x , x) ∈ K = cone



 epi

f t∗

+ epi σC ,

t∈T ) ∗ ∗ ∗ and ∃ λ ∈ R(T + , x t ∈ dom f t , αt ≥ 0, ∀t ∈ T , z ∈ dom σC , β ≥ 0, satisfying

−(x ∗ , x ∗ , x) =

(



λt xt∗ , f t∗ xt∗ + αt + z ∗ , σC z ∗ + β , t∈T

leading to (51) thanks to (4). The opposite implication is straightforward (standard arguments). At this point, we proceed by introducing a weaker constraint qualification. Previously, given z ∈ F and the set of indices corresponding to the active constraints at z, A(z) := {t ∈ T : f t (z) = 0}, it is easily verified that NC (z) + cone

) t∈A(z)

* ∂ f t (z) ⊆ NF (z).

(53)

If this inclusion is an equality, we say that the constraint system S is locally FarkasMinkowski (LFM, in short) at z ∈ F, and S is said to be LFM if it is LFM at every feasible point z ∈ F. In LSIP (C = Rn , f t (x) = at , x − bt , t ∈ T ), S is LFM at z ∈ F if NF (z) = cone {at , t ∈ T (z)} . The LFM property is closely related to the so-called basic constraint qualification (BCQ), which appeared in Hiriart-Urruty and Lemaréchal [28] for the ordinary convex programming problem, with equality and inequality constraints. In fact, LFM and BCQ are equivalent under the continuity of the function f := supt∈T f t at the reference point z and z ∈ int C. The LFM property was extended in [46] to the setting of linear semi-infinite systems. The consequences of its extension to convex semi-infinite systems were analyzed in [22]. For a deep analysis of BCQ and related conditions see also [37, 38]. An extensive comparative analysis of constraints qualifications for (P) is also given in [40].

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The following result is a LFM counterpart of a similar property for FM systems: If S is LFM at z ∈ F and for certain a ∗ ∈ X ∗ we have a ∗ , x ≤ a ∗ , z , for all x ∈ F, then a ∗ , x ≤ a ∗ , z is also a consequence of a finite subsystem of S. The converse statement holds provided that S is linear, but not in general for convex systems without any additional assumption. Obviously, if S is FM then it is LFM at any z ∈ F. Given the problem (P) and x ∈ F ∩ dom g, assuming again that g is continuous at some point of F and that S is LFM at x, the KKT conditions (51) also characterize the optimality of x. This conclusion comes straightforwardly from the PshenichnyiRockafellar theorem (e.g. [54, Theorem 2.9.1]) and the own definition of the LFM property: x is optimal for (P) ⇔ ∂g(x) ∩ (−NF (x)) = ∅ ⇔ θ ∈ ∂g(x) + NF (x) ( LFM ⇔ θ ∈ ∂g(x) + λt ∂ f t (x) + NC (x). t∈T (x)

We conclude this section by presenting the following KKT asymptotic optimality conditions. Given the problem (P), we assume that S is LFM and (ri F) ∩ dom g = ∅. Then, x ∈ F is a minimum of (P) if and only if, for each ε > 0 and a neighborhood ) of θ in X ∗ , U, there exists λ = λ(ε, U ) ∈ R(T + such that supp λ ⊂ A(x) := {t ∈ T : f t (x) = 0} and θ ∈ ∂ε g(x) +

#

t∈supp λ λt ∂ f t (x)

+ NC (x) + U.

(54)

The implication “⇒” is a direct consequence of (44) which establishes that, since (dom g) ∩ rint(F) = ∅, ∂(g + IF )(x) =



cl(∂gε (x) + NF (x)).

ε>0

Then, x is optimal for (P) ⇔ θ ∈

 ε>0

cl(∂gε (x) + NF (x),

so, for every given ε > 0 and any neighborhood of θ in X ∗ , U, θ ∈ ∂ε g(x) + N F (x) + U. Thus, by the LFM property, we have that  θ ∈ ∂ε g(x) + cone



 ∂ f t (x) ¯ + NC (x) ¯ + U,

t∈A(x) ¯

and we are done with the necessity statement. In order to prove the converse implication, fix x ∈ F (⊂ C). Given ε > 0, we choose a neighborhood of θ in X ∗ , U, such that |u ∗ , x − x| ≤ ε for all u ∗ ∈ U. If (54) holds, then there exists u ∗ε ∈ U such that

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u ∗ε ∈ ∂ε g(x) +

#

t∈supp λ λt ∂ f t (x)

) * # + NC (x) ⊂ ∂ε g + t∈supp λ λt f t + IC (x),

and we deduce g(x) +

#

t∈supp λ λt f t (x)

≥ g(x) +

#

t∈supp λ λt f t (x)

  + u ∗ε , x − x − ε.

Hence, since supp λε ⊂ A(x), g(x) ≥ g(x) +

#

t∈supp λ λt f t (x)

  ≥ g(x) + u ∗ε , x − x − ε ≥ g(x) − 2ε,

the desired conclusion follows by taking limits for ε → 0.

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Optimality Conditions for Quasi Proper Solutions in Multiobjective Optimization with a Polyhedral Cone L. Huerga , B. Jiménez , and V. Novo

Abstract In this work, a multiobjective optimization problem is considered, where the feasible set is given by a cone-constraint and the ordering cone is assumed to be polyhedral. Then, optimality conditions to characterize a type of quasi proper efficient solutions are obtained through linear scalarization and under generalized convexity hypotheses. These optimality conditions involve both the objective and the constraint functions and are given in terms of the classical subdifferential of Convex Analysis. Moreover, the scalarizing functional has an explicit and easy to handle expression, that depends on the matrix that defines the polyhedral ordering cone. Keywords Multiobjective optimization · Quasi proper efficiency · Polyhedral cone · Subdifferential

1 Introduction The notions of quasi efficiency appear in the literature with the aim of providing a set of points that are close to exact efficient solutions. This type of solution can be understood as an approximate solution in which the precision error varies with respect to the feasible points. The idea of its definition is very close to the Ekeland variational principle [1], that relates approximate solutions of a scalar optimization problem with exact solutions L. Huerga (B) · B. Jiménez · V. Novo Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), Calle Juan del Rosal, 12, 28040 Madrid, Spain e-mail: [email protected] B. Jiménez e-mail: [email protected] V. Novo e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_9

195

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of a perturbed problem, where the perturbation is given in terms of the distance function. The first concept of quasi efficiency for multiobjective optimization problems was introduced by Loridan [16], by considering the Pareto order. Since then, several papers more on this topic have been published for mutiobjective and general vector optimization problems (see, for instance [2, 3, 9, 10, 12, 13, 17, 18]). In particular, in [13] (see also [10]) the notion of quasi efficiency is extended to more general concepts of quasi efficiency/weak efficiency/proper efficiency for multiobjective optimization problems. In this work, we are going to focus on the notion of quasi proper efficiency introduced in [13], which combines the idea of quasi efficiency with the concept of proper efficiency in the sense of Henig [11]. These type of solutions are very interesting since they can be characterized through linear and nonlinear scalarization (see [13]). More precisely, we are going to consider a constrained multiobjective optimization problem, where the order relation is given by a polyhedral ordering cone, and the feasible set is given by a cone-constraint, and the aim is to continue the study initiated in [13] and provide alternative optimality conditions for quasi proper efficient solutions through linear scalarization under generalized convexity hypotheses. These optimality conditions will be stated by means of the classical subdifferential of Convex Analysis and involve both the objective and the constraint functions. It is worth to mention that the scalarizing functional is related to some special polyhedral cone, which is a perturbation of the original one. This fact lets us express these functionals in a explicit and algebraically way, in terms of the matrix that defines the ordering cone, which is very interesting from a computational point of view. The chapter is structured as follows. In Sect. 2, we state the framework, the notations and the main definitions and properties of quasi efficiency. Then, in Sect. 3, we establish the main result, in which we characterize quasi proper efficient solutions through linear scalarization in terms of optimality conditions of Karush-Kuhn-Tucher type. Finally, we state the conclusions.

2 Preliminaries Given a nonempty set F ⊂ Rn , we denote by int F, cl F, bd F, F c and cone F the topological interior, the closure, the boundary, the complementary and the cone  generated by F, respectively, where we remind that cone F = α≥0 αF. We say that F is a cone if αF ⊂ F, for all α ≥ 0. When int F = ∅, we say that F is solid. The set of nonnegative scalars is denoted as R+ and we denote R := R ∪ {+∞}. The Euclidean unit closed ball in Rn is denoted by B. In Rn , we consider a binary relation defined by a cone D as usual y1 ≤ D y2 ⇐⇒ y2 − y1 ∈ D, ∀y1 , y2 ∈ Rn .

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Also, when D is solid we define y1 < D y2 ⇐⇒ y2 − y1 ∈ int D, ∀y1 , y2 ∈ Rn . The positive and the strict positive polar cones of D are denoted, respectively, by D + and D s+ , that is, D + = {μ ∈ Rn : μ, d ≥ 0, ∀d ∈ D}, D s+ = {μ ∈ Rn : μ, d > 0, ∀d ∈ D\{0}}. In what follows, we suppose that D is polyhedral, i.e., it is defined as a finite intersection of half spaces p

D = {y ∈ Rn : Ay ∈ R+ },

(1)

where A ∈ M p×n (it is a matrix with p rows and n columns), p ≥ n. Then, from the definition, we deduce that D is closed and convex. Also, we assume that A has no zero rows and that it has full rank, which ensures that D is pointed (D ∩ (−D) = {0}). Thus, the binary relation in Rn induced by D is in fact a partial order. The problem that we are considering is the following Min{ f (x) : x ∈ Q},

(MOP)

n

where f = ( f 1 , f 2 , . . . , f n ) : Rm → R is the objective mapping and ∅ = Q ⊂ Rm is the feasible set. We suppose that Q ⊂ dom f := {x ∈ Rm : f (x) ∈ Rn }. We recall standard definitions of exact efficiency for problem (MOP) (see, for instance, [11, 14, 19]). A point x0 ∈ Q is said to be (i) an efficient solution of (MOP) if there is no x ∈ Q such that f (x) ≤ D f (x0 ), f (x) = f (x0 ); (ii) a weak efficient solution of (MOP) if int D = ∅ and there is no x ∈ Q such that f (x) < D f (x0 ); (iii) a proper efficient solution of (MOP) in the sense of Henig if there exists a proper (i.e., different from Rn ), solid and convex cone D  ⊂ Rn such that D\{0} ⊂ int D  and there is no x ∈ Q such that f (x) < D f (x0 ). If we denote the sets of efficient, weak efficient and proper efficient solutions of (MOP), respectively, by E( f, Q, D), WE( f, Q, D) and He( f, Q, D), it is clear that He( f, Q, D) ⊂ E( f, Q, D) ⊂ WE( f, Q, D). However, it is known that, frequently, the algorithms employed to solve an optimization problem deliver as solution points that are expected to be close to the efficient set. This fact makes necessary to study suitable notions of approximate efficiency.

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Related to the above, in [13], the authors introduced new notions of quasi efficiency/weak efficiency/proper efficiency for problem (MOP) that generalize and unify the most known concepts of approximate and quasi efficiency given in the literature (see also [10, 16] and the references therein). We remind them in the next definition. From now on, we consider a function h : Rm × Rm → R+ such that h(x, z) ≥ 0, for all x, z ∈ Rm , h(x, z) > 0 whenever x = z, and we introduce the following sets H := {∅ = C ⊂ Rn \{0} : C ∩ (−D) = ∅}, H := {∅ = C ⊂ Rn \{0} : cl coneC ∩ (−D\{0}) = ∅} ⊂ H,    D ⊂ Rn solid pointed convex cone such that . G(C) := D\{0} ⊂ int D  and C ∩ (− int D  ) = ∅ Definition 1 Let x0 ∈ Q and C ∈ H. (i) It is said that x0 is a (C, h)-quasi efficient solution of (MOP) if there is no x ∈ Q\{x0 } such that f (x0 ) ∈ f (x) + h(x, x0 )C. (ii) If C is solid, it is said that x0 is a (C, h)-quasi weak efficient solution of (MOP) if there is no x ∈ Q\{x0 } such that f (x0 ) ∈ f (x) + h(x, x0 ) int C. (iii) Suppose that C ∈ H. It is said that x0 is a (C, h)-quasi proper efficient solution of (MOP) if one can find some D  ∈ G(C) such that there is no x ∈ Q\{x0 } such that f (x0 ) ∈ f (x) + h(x, x0 )(C + int D  ). The set of (C, h)-quasi efficient, (C, h)-quasi weak efficient and (C, h)-quasi proper efficient solutions of (MOP) are denoted, respectively, by QE( f, Q, C, h), QWE( f, Q, C, h) and QPE( f, Q, C, h). Remark 1 (a) Let C ∈ H be given. By [13, Proposition 1] we have that QE( f, Q, C, h) ⊂ QWE( f, Q, C, h), whenever C is solid, and if C ∈ H it follows that  QE( f, Q, C + int D  , h) ⊂ QE( f, Q, C + D\{0}, h). QPE( f, Q, C, h) = D  ∈G(C)

(2) Also, if C ∈ H and D  ∈ G(C) it is deduced that

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QPE( f, Q, C + D  , h) = QPE( f, Q, C + int D  , h) = QE( f, Q, C + int D  , h). (3) (b) When C = D\{0}, the notions of quasi efficiency given in Definition 1 reduce to the respective concepts of exact efficiency. Besides, if h(x, z) = ε > 0, for all x, z ∈ Rm , the notions of quasi efficiency reduce to the respective concepts of (C, ε)efficiency/ -weak efficiency/ -proper efficiency introduced in [5, 8]. These notions generalize and unify the most important concepts of approximate efficiency with a fixed precision error given up to now. (c) If D = Rn+ , C = q + D, with q ∈ Rn+ and h(x, z) = x − z, given  ·  any norm in Rm , then the notion of quasi efficiency reduces to the definition due to Loridan [16]. (d) Finally, the concepts given in Definition 1 also extend the notions introduced in [10], in the sense indicated in [13, Proposition 2]. Thus, the (C, h)-quasi solutions must be understood as a type of approximate solutions in which the precision error is not fixed, but it changes depending on every feasible point. The selection of a convenient function h and a convenient set C to obtain a set of (C, h)-quasi efficient (respectively, quasi weak, quasi proper) solutions that approximates suitably the exact efficient set (respectively, weak efficient, proper efficient set) is the base to define an appropriate notion of approximate/quasi efficiency. A study in this direction can be found in papers [6, 7, 13]. Below we give more details. From now on, we are going to focus on (C, h)-quasi proper solutions of (MOP). These solutions are of special interest since they can be characterized through linear and nonlinear scalarization (see [13]). In this work, we are interested on continuing the study of optimality conditions for (C, h)-quasi proper solutions through linear scalarization processes. Section 3 is dedicated to this aim. Also, concerning to the set C, we are going to consider the sets C defined in the following way: C = q + D  , and C = q + D, for q ∈ D\{0} and D  ∈ G(D\{0}).

(4)

Note that family G(D\{0}) is formed by dilating cones for D (i.e., convex and pointed cones that contain D\{0} in their interior). These sets C are a good choice to obtain a set of (C, h)-quasi proper solutions and (C, ε)-proper solutions that provide a good approximation to the set of exact efficient solutions, as it can be seen in [13, Theorem 3], [6, Theorem 3] and [7, Theorem 4.6]. We indicate below [13, Theorem 3(a)], given for the particular case of (C, h)-quasi proper efficiency with respect to the sets C defined in (4) (take into account (3)). Theorem 1 Let x0 ∈ Q, q ∈ D\{0} and D  ∈ G(D\{0}). Let h n : Rm × Rm → R+ , n ∈ N, with h n (x, z) ≥ 0, for all x, z ∈ Rm , h n (x, z) > 0, whenever x = z, and let (xn ) ⊂ Q. If xn ∈ QPE( f, Q, q + D  , h n ), for all n ∈ N, f (xn ) → f (x0 ) and h n (·, xn ) → 0, then x0 ∈ He( f, Q, D).

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However, the family G(D\{0}) is, in general, very wide, and has a very abstract expression. To overcome this drawback, we are going to consider the following family of polyhedral dilating cones {Dρ , ρ > 0}, defined by Kaliszewski [15] for the polyhedral cone D, given by (1): p

Dρ := {y ∈ Rn : Ay + ρU Ay ∈ R+ }, being U ∈ M p× p the all-ones matrix. It follows that D0 = D, and Dρ \{0} ⊂ int Dρ , for all 0 ≤ ρ < ρ . Hence, Dρ ∈ G(D\{0}), for all ρ > 0. By [15, Lemma 3.7] we have that for any D  ∈ G(D\{0}), there exists ρ > 0 such that Dρ \{0} ⊂ int D  . Since the polyhedral cones are defined in terms of a matrix, the optimality conditions for a general optimization problem that involves such a type of cones are more interesting from a computational point of view. Thus, in what follows we fix q ∈ D\{0} and we consider the sets Cρ := q + Dρ , for ρ ≥ 0. If ρ = 0, we denote C := C0 . Note that these sets Cρ are much easier to compute. Example 1 Let m = n = 2, D = R2+ , f (x1 , x2 ) = (x1 , x2 ), for all (x1 , x2 ) ∈ R2 , and Q = R2+ . It follows that E( f, Q, D) = He( f, Q, D) = {(0, 0)} and WE( f, Q, D) = bd R2+ . Now consider q = (0, 1), ρ = 21 and let h(x1 , x2 ) = εx1 − x2 2 (the Euclidean norm), for 0 < ε < 1/3. It follows that QWE( f, Q, Cρ , h) = QE( f, Q, Cρ , h) = QPE( f, Q, Cρ , h) = {(0, 0)}. Thus, in this case, we obtain the exact efficient set. But, for instance, if we consider now that h(x1 , x2 ) = 1, for all (x1 , x2 ) ∈ R2 , then the set of (C, ε)-proper efficient solutions is given by   1 QPE( f, Q, Cρ , ε) = (x1 , x2 ) ∈ R2+ : x2 ≤ − x1 + ε . 3 which for ε > 0 small enough provides a bounded set of approximate proper solutions that approximates suitably the efficient set.

3 Optimality Conditions This section is devoted to derive optimality conditions for (C, h)-quasi proper solutions of problem (MOP) through linear scalarization, by means of the classical subdifferential of Convex Analysis.

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In what follows, we suppose that h satisfies the diagonal null property, i.e., h(x, x) = 0, for all x ∈ Rm . This property is verified, in particular, for the standard function h(x, z) = εx − z, with ε > 0, used commonly for this type of solutions, being  ·  any norm on Rm . Also, we are going to suppose that the feasible set Q of problem (MOP) is defined as Q := S ∩ M, where ∅ = M ⊂ Rm is a closed and convex set and S := {x ∈ Rm : g(x) ∈ −K }, where g = (g1 , g2 , . . . , gt ), g j : Rm → R, for j = 1, 2, . . . , t, and K ⊂ Rt is a cone, which is also assumed to be polyhedral, defined by K := {z ∈ Rt : Bz ∈ Rl+ }, being B ∈ Ml×t . In particular, when D = Rn+ (that is, when we consider the Pareto order) and K = Rt+ , we obtain the classical multiobjective optimization problem with inequality constraints. We need to introduce some more preliminaries to derive the results of this section. is nearly D-subconvexlike on Q if We say that a set-valued mapping G : Rm ⇒ Rn cl cone(G(Q) + D) is convex, where G(Q) := x∈Q G(x). We say that f is D-convex if dom f is convex and f (λx1 + (1 − λ)x2 ) ≤ D λ f (x1 ) + (1 − λ) f (x2 ), ∀x1 , x2 ∈ dom f, ∀λ ∈ (0, 1). Let us observe that f is D-convex if and only if ai ◦ f is convex, where ai refers to the ith row of matrix A, for all i = 1, 2, . . . , p. Also, for a function ϕ : Rm → R, we denote argmin Q ϕ := {z ∈ Q : ϕ(z) ≤ ϕ(x), ∀x ∈ Q}. We remind that the classical subdifferential of Convex Analysis of a function ϕ : Rm → R at a point x0 ∈ dom ϕ is defined as ∂ϕ(x0 ) := {x ∗ ∈ Rm : ϕ(x) ≥ ϕ(x0 ) + x ∗ , x − x0 , ∀x ∈ Rm }, and the normal cone of a nonempty closed convex set F ⊂ Rm at a point x0 ∈ F is given by N (F, x0 ) := {x ∗ ∈ Rm : x ∗ , x − x0  ≤ 0, ∀x ∈ F}. It follows that N (F, x0 ) = ∂ I F (x0 ), where I F denotes the indicator function of F (i.e., I F (x) = 0, if x ∈ F; I F (x) = +∞ otherwise). Besides, we consider the following Slater-type constraint qualification:

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(CQ) int dom f = ∅ and there exists x¯ ∈ M such that g(x) ¯ ∈ − int K . To establish our main result, we need the following theorem, which was proved in [13, Corollary 3] (and taking into account (3)). Theorem 2 Let ρ ≥ 0. (a) Suppose that ρ > 0 and the mapping x → f (x) − f (x0 ) + h(x, x0 )Cρ is nearly D-subconvexlike on Q for all x0 ∈ Q. Then, 

QPE( f, Q, Cρ , h) =

argmin Q (ξ ◦ f + h(·, x0 ) ξ, q).

ξ∈Dρ+ \{0}

(b) (ρ = 0) Suppose that the mapping x → f (x) − f (x0 ) + h(x, x0 )C is nearly D-subconvexlike on Q for all x0 ∈ Q. Then, QPE( f, Q, C, h) =



argmin Q (ξ ◦ f + h(·, x0 ) ξ, q).

ξ∈D s+

Remark 2 (a) In order to apply Theorem 2, let us note that for any ρ ≥ 0, we deduce that   p

 p p    λi ai + ρ λi ai : λ1 , λ2 , . . . , λ p ≥ 0 . Dρ+ = ξ ∈ Rn : ξ = i=1

i=1

i=1

(5) The strict positive polar cone Dρs+ has the same definition but considering in this case the scalars λi to be strictly positive, for all i = 1, 2, . . . , p. (b) In particular, if D = Rn+ and ρ > 0, taking into account (5), it is not difficult \{0} if and only if ξ ∈ int Rn+ and to check that ξ ∈ (Rn+ )+ ρ ξ−

ρ

ξ, ee ∈ Rn+ , 1 + nρ

where e denotes the all-ones vector in Rn . Now, we can state the main result. Theorem 3 Let x0 ∈ Q. Suppose that f is D-convex, g is K -convex, h(·, x0 ) is convex and the constraint qualification (CQ) holds. (a) Let ρ > 0. It follows that x0 ∈ QPE( f, Q, Cρ , h) if and only if there exist λi > 0, p i = 1, 2, . . . , p, with ξ := i=1 λi ai ∈ Dρ+ \{0} and ξ, q = 1, and γ j ≥ 0, for j = 1, 2, . . . , l, such that

Optimality Conditions for Quasi Proper …

0∈

p 

λi ∂(ai ◦ f )(x0 ) +

i=1

l 

203

γ j ∂(b j ◦ g)(x0 ) + ∂h(·, x0 )(x0 ) + N (M, x0 ),

j=1

(6) γ j (b j ◦ g)(x0 ) = 0, ∀ j = 1, 2, . . . , l.

(7)

(b) It follows that x0 ∈ QPE(  p f, Q, C, h) if and only if there exist λi > 0, i = 1, 2, . . . , p, with ξ := i=1 λi ai ∈ D s+ and ξ, q = 1, and γ j ≥ 0, for j = 1, 2, . . . , l, such that (6) and (7) are satisfied. Proof We only prove case (a), since the other one follows analogously. Let x0 ∈ QPE( f, Q, Cρ , h). We observe that mapping x → f (x) − f (x0 ) + h(x, x0 )(q + Dρ ) can be expressed equivalently as x → p(x) + Dρ , where p(x) := ( f + h(·, x0 )q)(x) − f (x0 ). Under the hypotheses, it is clear that p is D-convex on Q, so by [4, Theorem 2.7(a)] (for C := Dρ , K := D, and for any ε ≥ 0 in this result) we have that cl cone( p(Q) + Dρ ) is convex, so mapping x → f (x) − f (x0 ) + h(x, x0 )(q + Dρ ) is nearly Dsubconvexlike on Q, for all x0 ∈ Q. Then, by Theorem 2(a), there exists κ ∈ Dρ+ \{0} such that x0 ∈ argmin Q (κ ◦ f + h(·, x0 ) κ, q). 1 κ∈ Since q ∈ D\{0} ⊂ int Dρ , it follows that κ, q > 0. Consider ξ := κ,q + Dρ \{0}. Then, we have that ξ, q = 1 and x0 ∈ argmin Q (ξ ◦ f + h(·, x0 )), which is equivalent to x0 ∈ argminRm (ξ ◦ f + h(·, x0 ) + I S + I M ), that is 0 ∈ ∂(ξ ◦ f + h(·, x0 ) + I S + I M )(x0 ).

(8)

Thus, under the given hypotheses, the sum rule for proper extended real-valued convex mappings is satisfied (see, for instance, [20, Theorem 2.8.7]), so (8) is equivalent to 0 ∈ ∂(ξ ◦ f )(x0 ) + ∂h(·, x0 )(x0 ) + ∂ I S (x0 ) + ∂ I M (x0 ).

(9)

On the other hand, since ξ ∈ Dρ+ \{0}, by statement (5) there exist μ1 , μ2 , . . . , μ p ≥ 0 such that ξ=

p  i=1

(μi + ρs)ai ,

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p where s = i=1 μi . Since ξ = 0, we deduce that s > 0, so λi := μi + ρs > 0, for all i = 1, 2, . . . , p. Besides, as f is D-convex, it follows that ai ◦ f is convex, and then continuous in int dom f for all i = 1, 2, . . . , p. Thus, we can apply again the sum rule for the subdifferential and we obtain that ∂(ξ ◦ f )(x0 ) =

p 

λi ∂(ai ◦ f )(x0 ).

(10)

i=1

Moreover, it is known that (see, for instance [20, Chap. XI, Corollary 3.6.2] and [6, Lemma 2]) there exist γ j ≥ 0, with γ j (b j ◦ g)(x0 ) = 0, for all j = 1, 2, . . . , l, such that ∂ I S (x0 ) = N (S, x0 ) =

l 

γ j ∂(b j ◦ g)(x0 ).

(11)

j=1

Then, statements (a) and (b) follow directly taking into account (9), (10) and (11).  pReciprocally,+ suppose that there exist λi > 0, for i = 1, 2, . . . , p, with ξ := i=1 λi ai ∈ Dρ \{0} and ξ, q = 1, and γ j ≥ 0, for j = 1, 2, . . . , l such that statements (a) and (b) hold. Then, there exist xi∗ ∈ ∂(ai ◦ f )(x0 ), for i = 1, 2, . . . , p, y ∗j ∈ ∂(b j ◦ g)(x0 ), for j = 1, 2, . . . , l, w ∗ ∈ ∂h(·, x0 )(x0 ) and z ∗ ∈ N (M, x0 ) such that 0=

p 

λi xi∗ +

i=1

l 

γ j y ∗j + w ∗ + z ∗ .

(12)

j=1

We have that λi (ai ◦ f )(x) ≥ λi (ai ◦ f )(x0 ) + λi xi∗ , x − x0 , ∀x ∈ Rm , ∀i = 1, 2, . . . , p, γ j (b j ◦ g)(x) ≥ γ j y ∗j , x − x0 , ∀x ∈ Rm , ∀ j = 1, 2, . . . , l, h(x, x0 ) ≥ w ∗ , x − x0 , ∀x ∈ Rm , 0 ≥ z ∗ , x − x0 , ∀x ∈ M. By adding all the inequalities above and taking into account that γ j (b j ◦ g)(x) ≤ 0, for all x ∈ S, we deduce that p 

λi (ai ◦ f )(x) + h(x, x0 ) ≥

i=1

p 

λi (ai ◦ f )(x0 )

i=1

+ λi xi∗ + γ j y ∗j + w ∗ + z ∗ , x − x0 , ∀x ∈ Q, which by (12) is equivalent to

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ξ, f (x) − f (x0 ) + h(x, x0 )q ≥ 0, ∀x ∈ Q.

(13)

Statement (13) is at the same time equivalent to saying that x0 ∈ argmin Q (ξ ◦ f + h(·, x0 ) ξ, q), and hence, by Theorem 2(a) we deduce that x0 ∈ QPE( f, Q, Cρ , h). The proof is complete. Remark 3 (a) Let us observe that in Theorem 3, all the components ai ◦ f , i = 1, 2, . . . , p, have an effective role in the optimality conditions for quasi proper solutions of problem (MOP). That is, we have derived a kind of strong KarushKuhn-Tucker optimality conditions for this type of solutions. (b) In particular, if h(x, z) = εx − z2 (the Euclidean norm), for all x, z ∈ Rm , with ε > 0, then we know that (see, for instance, [20, Corollary 2.4.16]) ∂h(·, x0 )(x0 ) = εB. In the next corollary, we establish optimality conditions for quasi proper solutions of problem (MOP) in the particular case when D = Rn+ , K = Rt+ and h(x, z) = εx − z2 , for all x, z ∈ Rm , with ε > 0. The proof follows straightforward from Theorem 3 and Remark 2(b). Corollary 1 Suppose that D = Rn+ , K = Rt+ and h(x, z) = εx − z2 , for all x, z ∈ Rm , with ε > 0. Let x0 ∈ Q. Suppose that f is D-convex, g is K -convex and (CQ) holds. (a) Let ρ > 0. It follows that x0 ∈ QPE( f, Q, Cρ , h) if and only if there exist ξ := e ∈ Rn+ and ξ, q = 1, and γ j ≥ 0, (λ1 , λ2 , . . . , λn ) ∈ int Rn+ , with ξ − ρ ξ,e 1+ρn for j = 1, 2, . . . , t, such that 0∈

n  i=1

λi ∂ f i (x0 ) +

t 

γ j ∂g j (x0 ) + εB + N (M, x0 ),

(14)

j=1

γ j g j (x0 ) = 0, ∀ j = 1, 2, . . . , t.

(15)

(b) We have that x0 ∈ QPE( f, Q, C, h) if and only if there are ξ := (λ1 , λ2 , . . . , λ p ) ∈ int Rn+ , with ξ, q = 1, and γ j ≥ 0, for j = 1, 2, . . . , t, such that (14) and (15) are verified. Remark 4 In this work we have focused on (Cρ , h)-quasi proper solutions for (MOP) when h satisfies the diagonal null property. Another case is when h(x, z) = ε > 0. We know then that in this case the notion of (C, h)-quasi proper efficiency reduces to the concept of (C, ε)-proper efficiency in the sense of Henig, introduced by Gutiérrez et al. in [5]. For this type of solutions, the same authors derived in [6] optimality conditions through linear scalarization, also in terms of the classical subdifferential of Convex Analysis. We illustrate the results in the following example.

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Fig. 1 Set f (Q)

Example 2 Let m = n = 2, M = R2 , D = R2+ , K = R2+ , f (x, y) = (x, (y − 1)2 ), g(x, y) = (x − y, −x − y), q = (1, 1) and ε = ρ = 21 . We suppose that h(x, z) = εx − z2 , for all x, z ∈ Rm . Then, it follows that Q = {(x, y) ∈ R2 : x ≤ y and − x ≤ y}. The image by f of the feasible set Q is represented in Fig. 1. We are going to find all the points (a, b) ∈ QPE( f, Q, Cρ , h). It is clear that, under the given data, the assumptions of Corollary 1 are satisfied. Then, by means of this result, we have to find the pairs (a, b) ∈ Q for which there exist λ1 , λ2 > 0 and γ1 , γ2 ≥ 0 such that the following conditions hold: ⎧ (0, 0) ∈ λ1 ∇ f 1 (a, b) + λ2 ∇ f 2 (a, b) + γ1 ∇g1 (a, b) + γ2 ∇g2 (a, b) + εB, ⎪ ⎪ ⎪ ⎪ γ1 g1 (a, b) = 0, ⎪ ⎪ ⎨ γ2 g2 (a, b) = 0, ρ(λ1 +λ2 )

λ1 − 1+2ρ ≥ 0, ⎪ ⎪ ⎪ 1 +λ2 ) ⎪ ⎪ ≥ 0, λ − ρ(λ1+2ρ ⎪ ⎩ 2 λ1 + λ2 = 1, or equivalently ⎧ (0, 0) ∈ λ1 (1, 0) + λ2 (0, 2(b − 1)) + γ1 (1, −1) + γ2 (−1, −1) + εB, ⎪ ⎪ ⎪ ⎪ γ1 (a − b) = 0, ⎪ ⎪ ⎨ γ2 (−a − b) = 0, λ1 ≥ 41 , ⎪ ⎪ ⎪ ⎪ λ ≥ 1, ⎪ ⎪ ⎩ 2 4 λ1 + λ2 = 1.

(16)

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The first equation is equivalent to (−λ1 − γ1 + γ2 , −2(b − 1)λ2 + γ1 + γ2 ) ∈ εB, or equivalently (λ1 + γ1 − γ2 )2 + [2(b − 1)λ2 − γ1 − γ2 ]2 ≤ ε2 .

(17)

The last three equations in (16) imply 1 1 3 3 ≤ λ1 ≤ and ≤ λ2 ≤ . 4 4 4 4

(18)

Case 1) Suppose first that (a, b) ∈ int Q, i.e., a = b and a = −b. Then γ1 = 0, γ2 = 0 and (17) becomes λ21 + 4(b − 1)2 λ22 ≤ ε2 . Replacing ε = 21 , λ2 = 1 − λ1 , we derive 4(b − 1)2 (1 − λ1 )2 ≤ lently  1 1 − λ21 . |b − 1| ≤ 2(1 − λ1 ) 4

1 4

− λ21 , or equiva-

There exists a solution b ∈ R if and only if 41 − λ21 ≥ 0, i.e., (using also (18)) 41 ≤ λ1 ≤ 21 . To find all the possible values of b we use the maximum value of the function  1 1 λ1 → 2(1−λ − λ21 with 41 ≤ λ1 ≤ 21 . The maximum is attained at λ1 = 41 , so 4 1) √

√

λ2 = 43 , and the maximum value is 63 . Thus |b − 1| ≤ 63 , and as (a, b) ∈ int Q we obtain √ √ 3 3 ≤b ≤1+ and − b < a < b. 1− 6 6 Case 2) a = b. Then γ1 ≥ 0, γ2 = 0 and, using that λ2 = 1 − λ1 , (17) becomes (λ1 + γ1 )2 + [2(b − 1)(1 − λ1 ) − γ1 ]2 ≤

1 . 4

From here we obtain G(λ1 , γ1 ) ≤ b − 1 ≤ F(λ1 , γ1 ),

(19)

where G(λ1 , γ1 ) :=

γ1 −



1 4

− (λ1 + γ1 )2

2(1 − λ1 )

and F(λ1 , γ1 ) :=

γ1 +



1 4

− (λ1 + γ1 )2

2(1 − λ1 )

.

For a solution to exist, it is necessary that λ1 + γ1 ≤ 1/2. Since λ1 ≥ 1/4 we obtain that γ1 ≤ 1/4. On the other hand, taking into account (19), we observe that we are looking for values b such that

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b−1∈

[G(λ1 , γ1 ), F(λ1 , γ1 )],

(λ1 ,γ1 )∈R

  where R := (λ1 , γ1 ) ∈ R2 : 14 ≤ λ1 ≤ 43 , λ1 + γ1 ≤ 21 , 0 ≤ γ1 ≤ 41 , which is the       triangle with vertices 41 , 41 , 41 , 0 and 21 , 0 . Thus, by studying the maximum of function F and the minimum of function G in R (which are attained, since F and G are continuous on R, and R is compact), we deduce that (b, b) ∈ QPE( f, Q, Cρ , h) if and only if 

 √ √ 2 2−1 3 ,1 + b ∈ 1− . 6 6 Case 3) a = −b. Then γ1 = 0, γ2 ≥ 0 and, using that λ2 = 1 − λ1 , (17) becomes (λ1 − γ2 )2 + [2(b − 1)(1 − λ1 ) − γ2 ]2 ≤ From here we obtain  γ2 − 41 − (λ1 − γ2 )2 2(1 − λ1 )

≤b−1≤

γ2 +



1 4

1 . 4

− (λ1 − γ2 )2

2(1 − λ1 )

,

and by reasoning in analogous way as in the second case, we deduce that (−b, b) ∈ QPE( f, Q, Cρ , h) if and only if 

√  √ 3+2 2 3 ,1 + b ∈ 1− . 6 2 In sum, we have that (a, b) ∈ QPE( f, Q, Cρ , h) if and only if √ 3 3 ≤b ≤1+ , − b < a < b and 1 − √ 6 √ 6 2 2−1 3 or a = b and 1 − ≤b ≤1+ , 6√ 6 √ 3+2 2 3 or a = −b and 1 − ≤b ≤1+ . 6 2 √

The set QPE( f, Q, Cρ , h) and its image are represented in Figs. 2 and 3, respectively.

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Fig. 2 Set QPE( f, Q, Cρ , h) (in purple) Fig. 3 Set f (QPE( f, Q, Cρ , h)) (in purple)

4 Conclusions In the setting of a constrained multiobjective optimization problem with a polyhedral ordering cone, optimality conditions to characterize quasi proper efficient solutions have been derived in terms of the classical subdifferential of Convex Analysis. These optimality conditions are given through linear scalarization and involve both the objective and the constraint functions. The related scalarizing functional depends on a polyhedral dilating cone of the ordering cone and has an explicit expression, which makes these optimality conditions more interesting from a computational point of view.

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As a future line of research, it would be interesting to complete this study and obtain alternative optimality conditions in terms of generalized weak subdifferentials, defined by means of the quasi proper efficiency notion, and by considering weaker convexity conditions. Acknowledgements This work is partially supported by Ministerio de Ciencia e Innovación and Agencia Estatal de Investigación (Spain) under project with reference PID2020-112491GBI00/AEI/10.13039/501100011033, and also by E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), Spain, under grant 2022-ETSII-17.

References 1. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (2014) 2. Govil, M.G., Mehra, A.: ε-optimality for multiobjective programming on a Banach space. European J. Oper. Res. 157, 106–112 (2004) 3. Gupta, D., Mehra, A.: Two types of approximate saddle points. Numer. Funct. Anal. Optim. 29, 532–550 (2008) 4. Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Proper approximate solutions and εsubdifferentials in vector optimization: basic properties and limit behaviour. Nonlinear Anal. 79, 52–67 (2013) 5. Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Henig approximate proper efficiency and optimization problems with difference of vector mappings. J. Convex. Anal. 23, 661–690 (2016) 6. Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Optimality conditions for approximate proper solutions in multiobjective optimization with polyhedral cones. TOP 28, 526–544 (2020) 7. Gutiérrez, C., Huerga, L., Novo, V., Sama, M.: Limit behaviour of approximate proper solutions in vector optimization. SIAM J. Optim. 29, 2677–2696 (2019) 8. Gutiérrez, C., Jiménez, B., Novo, V.: On approximate efficiency in multiobjective programming. Math. Methods Oper. Res. 64, 165–185 (2006) 9. Gutiérrez, C., Jiménez, B., Novo, V.: Optimality conditions for quasi-solutions of vector optimization problems. J. Optim. Theory Appl. 167, 796–820 (2015) 10. Gutiérrez, C., López, R., Novo, V.: Generalized ε-quasi-solutions in multiobjective optimization problems: existence results and optimality conditions. Nonlinear Anal. 72, 4331–4346 (2010) 11. Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982) 12. Huang, X.X.: Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization. Optimization 51, 309–321 (2002) 13. Huerga, L., Jiménez, B., Luc, D. T., Novo, V.: A unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimization. Math. Program. 189(1–2), Ser. B, 379–407 (2021) 14. Jahn, J.: Vector Optimization. Theory, Applications, and Extensions, Springer, Berlin (2011) 15. Kaliszewski, I.: Quantitative Pareto Analysis by Cone Separation Technique. Kluwer Academic Publishers, Boston (1994) 16. Loridan, P.: ε-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984) 17. Luc, D. T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989) 18. Luc, D.T., Ngai, D.T., Théra, M.: Approximate convex functions. J. Nonlinear Convex Anal. 1, 155–176 (2000)

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19. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, Orlando (1985) 20. Z˘alinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

On Distinguished Spaces C p (X) of Continuous Functions J. K¸akol and A. Leiderman

Abstract This survey paper presents a number of recent results related to the study of spaces C p (X ) of continuous real-valued functions on a Tychonoff spaces X with the pointwise topology that are distinguished, i.e. spaces C p (X ) which are large subspaces of R X , equivalently, spaces C p (X ) whose strong dual L β (X ) of C p (X ) carries the finest locally convex topology. Some open questions are included. Keywords Distinguished space · Bidual space · Eberlein compact space · Fré chet space · Strongly splittable space · Q-space · Point-finite family 2020 Mathematics Subject Classification 54C35 · 46A03

1 Introduction Recall that a locally convex space E (embedded in its bidual E  by means of the evaluation map) is semi-reflexive if E  coincides algebraically with E, reflexive if it is semi-reflexive and the original locally convex topology of E coincides with  β E  , E  , and distinguished if its strong dual E β is barrelled. It is obvious that each reflexive space is semi-reflexive, and each semi-reflexive space is distinguished [30, 23.3 (4)]. In fact, (alternative definition [30, 23.7]), a ˇ Project 20-22230L and RVO: The research of the first named author is supported by the GACR 67985840. He thanks also the Center For Advanced Studies in Mathematics of Ben-Gurion University of the Negev for financial support during his visit in 2022. J. K¸akol (B) Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614, Pozna´n, Poland Institute of Mathematics, Czech Academy of Sciences, Prague, Czechia e-mail: [email protected] A. Leiderman Department of Mathematics, Ben-Gurion University of Negev, Beer Sheva, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_10

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convex space E is distinguished if and only if E is a large subspace of locally E  , σ E  , E  . Recall also that a subspace F of a locally convex space G is a large subspace of G if for every bounded set B ⊂ G there is a bounded set A ⊂ F such that B ⊂ A, where the closure is taken in G [38, Definition 8.3.22]. If X is a Tychonoff space and C p (X ) denotes the linear space C(X ) of all realvalued continuous functions defined on X equipped with the pointwise topology, it can be easily seen that C p (X ) is semi-reflexive if and only if C p (X ) is reflexive, if and only if X is discrete. The first statement follows directly from [30, 23.5 (1)] and [25, 11.3 Corollary]. The second one is a consequence of [12, Corollary 3.4]. The simplest examples of distinguished C p (X ) spaces which are not semireflexive are those with X any countable nondiscrete Tychonoff space (see [12] and Theorem 2.5 below). This survey article put together several recently obtained results about distinguished spaces C p (X ). Most of presented here results about such spaces C p (X ) have been published in articles [12, 15, 17, 28, 29, 33, 34]. It is shown in [15, Theorem 10] that C p (X ) is distinguished if and only if C p (X ) is a large subspace of R X . Equivalently, C p (X ) is distinguished if and only if its strong dual carries the finest locally convex topology. A characterization of distinguished C p (X ) spaces in terms of X has been obtained independently in [15, Theorem 5] and [28, Theorem 2.1]. In [28, Theorem 2.1] we proved that a locally convex space C p (X ) is distinguished if and only if X is a -space. This theorem apparently provides (with several applications) a significant “bridge” between attractive problems from the set theory and related with -sets, λ-sets, and Q-sets X , respectively, and corresponding distinguished spaces C p (X ). In the following chapters of this work, a number of results concerning the properties of -spaces X and their applications for the study of the space C p (X ) will be presented. In this paper we shall assume that all linear spaces are over the field R of real numbers and all locally convex spaces are Hausdorff. The topological dual of C p (X ) will be denoted by L(X ), or by L p (X ) when equipped with the weak* topology. We shall designate by Ck (X ) the space C(X ) equipped with the compact-open topology.

2 General Facts About Distinguished Spaces Distinguished locally convex spaces were introduced by J. Dieudonné and L. Schwartz. A. Grothendieck showed that a metrizable locally convex space E is distinguished if and only if its strong dual E β = (E  , β(E  , E)) is bornological. Heinrich [21] proved that each quasinormable locally convex space [38, Definition 8.3.34] satisfies the density condition; what applies to show that every metrizable quasinormable locally convex space is distinguished, see [7].

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A very important and applicable fact related with distinguished Fréchet spaces (i.e. metrizable and complete locally convex spaces) asserts the following: A Fréchet space E is distinguished if and only if the strong dual of E is a regular (L B) -space, see [38, Observation 8.5.14 (e)]. We refer to [24, 3.16], [25, 13.4], [30, 23.7, 29.3] and [38, 8.3] for further information about distinguished metrizable locally convex spaces. On the other hand, a Fréchet space E is distinguished if and only if its strong dual E β has countable tightness [14, Corolary 4]. The most celebrated example of a nondistinguished Fréchet space is Köthe’s echelon space λ [T], [30, 31.7]. We refer the reader to [47] for other examples of non-distinguished Fréchet spaces. Recall that a locally convex space E is called quasinormable [38, Definition 8.3.34] if for every absolutely convex neighbourhood of zero U in E there exists an absolutely convex neighbourhood of zero V ⊆ U such that for every λ > 0 there exists a bounded set B in E with V ⊆ B + λU . The class of quasinormable spaces contains, for example, the (D F)-spaces, the metrizable Ck (X ) spaces, the spaces C n () for n ∈ ω,  being an open subset of Rω , as well as all Fréchet-Montel spaces (see [25, 37]). A metrizable locally convex space E with a decreasing base {Un : n ∈ ω} of absolutely convex neighbourhoods of zero satisfies the density condition if there is  a double sequence Bn,k : n, k ∈ ω of bounded sets in E such that for each n ∈ ω and each bounded set C ⊆ E there is k ∈ ω with C ⊆ Bn,k + Un , [8, Theorem 9]. We recall also the following general result characterizing the density condition for metrizable locally convex spaces. Theorem 2.1 ([8]) For a metrizable locally convex space E the following assertions are equivalent. (1) E satisfies the density condition. (2) Every bounded set in E β is metrizable. (3) The space 1 (E) is distinguished. On the other hand, the following result from [13] supplements the above theorem. Theorem 2.2 ([13]) A metrizable locally convex space E is distinguished if and only if every bounded set in the strong dual of E has countable tightness. Consequently, if E is a metrizable locally convex space that satisfies the density condition, Theorem 2.1 ensures that every bounded set in the strong dual of E is metrizable, which allows Theorem 2.2 to guarantee that E is distinguished. The converse fails, see [9]. We also refer the reader to [8], where some results about distinguished Köthe echelon and co-echelon spaces are presented. Since, as mentioned earlier, each Ck (X ) space is quasinormable (see [25, 10.8.2 Theorem]), and metrizable quasinormable spaces are distinguished, we have the following.

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Proposition 2.3 Every metrizable Ck (X ) space is distinguished. A similar result also holds for spaces C p (X ). If Q ⊆ L (X ) has an infinite support S = {supp μ : μ ∈ Q}, there are a discrete sequence {xn }∞ n=1 in X contained in S, a ∞ } { in Q and a bounded sequence f in C sequence {μn }∞ n p (X ) with x n ∈ supp μn n=1 n=1 and μn , f n  = n for all n ∈ N, [25, 11.7.2 Theorem]. The proof of the first statement of the next result is similar to that of [25, 10.8.2 Theorem]. Note that the second statement can be found in [25, 11.7.3 Corollary]. Theorem 2.4 The space C p (X ) is always a quasinormable and quasibarrelled space. Proof Sketch of the proofs. For a finite A ⊆ X , and  > 0, set  U=

 f ∈ C (X ) : sup | f (x)| <  . x∈A

  With B := f ∈ C (X ) : supx∈X | f (x)| ≤ 2 , a similar argument to the proof of [25, 10.8.2 Theorem] shows that U ⊆ B + λU for 0 < λ ≤ 1, so for all λ > 0 since U is absolutely convex. For the second statement, if Q is a strongly bounded set in L (X ), the above obser vation implies that {supp μ : μ ∈ Q} is finite. So Q belongs to a finite-dimensional subspace of L (X ). This means that the polar Q 0 of Q in C (X ) is a neighbourhood of zero in C p (X ).  Since every metrizable quasinormable locally convex space is distinguished (as mentioned above, due to a result of Heinrich), we have the following consequence of Theorem 2.4. Theorem 2.5 ([12, Theorem 3.3 (a)]) For a countable Tychonoff space X the space C p (X ) is distinguished. In the sequel we develop another purely topological approach to the proof of Theorem 2.5. To provide a characterization of distinguished spaces C p (X ) we start with the following general fact (which will be used below) describing those locally convex spaces which carry the strongest locally convex topology, see [15]. Theorem 2.6 The following assertions are equivalent for a locally convex space E. (1) (2) (3) (4)

E has its strongest locally convex topology. Every absolutely convex absorbing subset of E is a neighbourhood of zero. E is the strong dual of the product of dim E-many lines. E is barrelled and admits a continuous basis [41].

Proof The equivalences between (1), (2) and (3) are elementary facts from the theory of topological vector spaces.

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(1) ⇒ (4): For the strongest locally topology ξ on a vector space E we know that every linear functional is ξ-continuous, so every basis is ξ-continuous. Moreover, the finest locally convex topology ξ of E is the direct sum topology of all its finite-dimensional locally convex subspaces, so it must be a barrelled space, see [38, Proposition 4.2.6]. (4) ⇒ (2): Let H ⊂ E be a Hamel basis of E with continuous coefficient functionals f x for each x ∈ H . Let V be an absolutely convex absorbing subset of E. For each x ∈ H there exists tx > 0 such that x ∗ = tx x ∈ V . Set A = {x ∗ : x ∈ H }. Then A is another Hamel basis of E with continuous coefficients gx∗ = (tx )−1 f x . The polar set A◦ is σ(E  , E)-bounded, so the bipolar A◦◦ is a neighbourhood of zero in the barrelled space E. Now let y=



ax x ∗ ∈ A◦◦

x∈U

for some finite subset U ⊂ H and scalars ax . Choose g=

(sgn(ax ))h x ∗ ∈ A◦ . x∈U

Then



|ax | = |g(y)| ≤ 1.

x∈U

This shows that A◦◦ ⊂ V . Hence V is a neighbourhood of zero in E as we wanted to show.  The character χ (E) of a locally convex space E is the smallest cardinality for a base of neighbourhoods of zero in E. A locally convex space E is feral if every bounded subset is finite-dimensional, flat if every linear functional over E is continuous, and fit if it has a dense (linear) subspace whose codimension equals the dimension of E (see [42, 44]). Each flat space is feral, see also [14] for detail. Theorem 2.7 ([42, Theorem 1]) A locally convex space E is fit if χ (E) ≤ dim E. Strong duals of metrizable spaces are (D F)-spaces. We contribute with the following result from [15, Theorem 7]. Theorem 2.8 ([15]) The strong dual of a metrizable locally convex space E is either fit or flat. Proof (1) Assume no infinite-dimensional subspace of E admits a continuous norm,   i. e., E has its weak topology σ E, E  . Then E is (isomorphically) a dense subspace of the product G of dim E  -many lines, and metrizability implies dim E  ≤ ℵ0 . Thus G is separable and metrizable, so E is a large subspace of G [38, 8.3.23 (b)] and has the same strong dual as G. Hence the strong dual E β of E has its strongest locally convex topology and therefore is flat (and barrelled).

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(2) Assume some infinite-dimensional subspace of E admits a continuous norm. Then dim E  ≥ c by the theorems of Mazur and Hahn-Banach. Metrizability and [43, Theorem 1] yield a fundamental family B of bounded sets in E with |B| ≤ d, with d the dominating cardinal. Therefore   χ E β ≤ d ≤ c ≤ dim E β , so that E β is fit by Theorem 2.7.



Part (1) of the proof provides also another simpler proof of Theorem 2.5; if X is countable, the space L β (X ) has its strongest locally convex topology and thus admits a continuous basis. It turns out, all L β (X ) spaces admit continuous bases. We recall here the following fundamental fact concerning the strong dual L β (X ), see [15]. Theorem 2.9 ([15, Theorem 7]) The homeomorphic copy of X in L p (X ) is a continuous basis for L β (X ). Proof The set A of functions g in C (X ) such that |g (x)| ≤ 1 for all x ∈ X is pointwise bounded, so the polar A0 is a neighbourhood of zero in L β (X ). Tychonoff 0 extension

theory ensures A is just the absolutely convex hull of X . To see this, let y := x∈ ax · x be a finite linear combination from X , choose g ∈ A with g (x) = sgnax for each x ∈ , and note that |g, y| =



|ax | ≤ 1

x∈

if and only if y ∈ A0 . Thus the coefficient functionals for the basis X are all bounded by 1 on A0 , so are continuous on L β (X ).  This theorem yields the following Theorem 2.10 ([15]) For a Tychonoff space X the following assertions are equivalent. (1) C p (X ) is distinguished. (2) C p (X ) is a large subspace of R X . (3) L β (X ) has its strongest locally convex topology, [12]. Proof Theorems 2.6 and 2.9 show the equivalence (1) ⇔ (3), which is just mentioned in [12, Corollary 3.4]. Since C p (X ) is a dense subspace of the product R X , we may algebraically identify L (X ) as their common dual by means of restriction. Then L β (X ) is their common strong dual if and only if C p (X ) is large in R X , by the bipolar theorem. Theorem 2.6 shows (2) ⇔ (3). 

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3 A Characterization of Distinguished Spaces C p (X) in the Frame of R X Most of results presented in this section were proved in [15]. By Theorem 2.10 we know already that C p (X ) is distinguished if and only if its strong dual carries the finest locally convex topology. The following result gathers together more equivalent conditions characterising distinguished spaces C p (X ). Theorem 3.1 ([15]) For a Tychonoff space X the following assertions are equivalent: (1) C p (X ) is distinguished. (2) The strong bidual of C p (X ) is the product space R X . (3) The strong bidual E of C p (X ) is quasi-complete, i.e. every closed bounded set in E is complete. (4) For each bounded set B ⊂ R X there exists a bounded set A ⊂ C p (X ) such that B ⊂ A, where the closure is taken in R X . (5) C p (X ) ⊂ R X is large, i.e. for each f ∈ R X there exists a bounded set A ⊂ C p (X ) such that f ∈ A. Last Theorem 3.1 ((5) ⇒ (1)) implies also the statement of Theorem 2.5 mentioned above: If X is a countable Tychonoff space then C p (X ) is distinguished. Recently, Ferrando [17] obtained necessary and sufficient conditions about how to approximate a function f ∈ R X by a pointwise bounded family in C(X ). This is closely related to the research on the distinguished property of C p (X ). The following proposition provides another possible constructions of getting distinguished spaces C p (X ). Proposition 3.2 ([15, Proposition 19]) Let {X i : i ∈ I } be a family of Tychonoff spaces and let ⊕i∈I X i denote their topological sum. If each C p (X i ) is distinguished, then C p (⊕i∈I X i ) is distinguished. Proof As the locally convex space C p (⊕i∈I X i ) is a (linearly homeomorphic) copy  of the product space i∈I C p (X i ), its strong dual is a copy of the direct sum  ⊕i∈I L β (X i ) of barrelled spaces. Proposition 3.3 The space C p (X ) is distinguished if and only if the subspace C bp (X ) of C p (X ) consisting of all bounded functions is distinguished. Proof Since each pointwise bounded set A in C (X ) is contained in the closure in C p (X ) of a pointwise bounded subset of C b (X ), one has that C bp (X ) is a large  subspace of C p (X ). So, both strong duals are isomorphic. We will also use in the sequel the following fact stating that Theorem 3.4 ([15, Theorem 16]) If Y ⊂ X is a subspace of a Tychonoff space X and C p (X ) is distinguished, then C p (Y ) is distinguished. Note that Propositions 3.2, 3.3 and Theorem 3.4 are simple consequences of Theorem 5.2.

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4 Distinguished Spaces C p (X) Over Uncountable Compact Spaces X Recall that the Lindelöfication L m of an uncountable discrete space D (m) is a nondiscrete P-space such that C p (L m ) is distinguished (see [12, Example 3.8] for m = ℵ1 ). We show below that there exist also distinguished spaces C p (X ) over uncountable compact X . First we recall that a family F of subsets of X is called point-finite [1] if each x ∈ X belongs at most to finitely many members of F. Let us introduce the following concept which has been used in [15] but under the name scant cover. Definition 4.1 ([15]) We say that a family {Nx : x ∈ X } of subsets of a Tychonoff space X is a point-finite open assignment for X if each Nx is a neighbourhood of x and for each u ∈ X the set X u = {x ∈ X : u ∈ Nx } is finite. Example 4.2 Each countable Tychonoff space X = {xn : n ∈ N} admits a pointfinite open assignment. Simply set Nxn = {xi : i ≥ n}for each n ∈ ω. Next theorem from [15] will be used below. Theorem 4.3 ([15, Theorem 46]) If X admits a point-finite open assignment {Nx : x ∈ X } then C p (X ) is distinguished. Last Theorem 4.3 applies to get the following Corollary 4.4 If X has only finitely many non-isolated points, then C p (X ) is distinguished. Proof If X =  ∪ {u 1 , . . . , u n }, where all points x ∈  are isolated in X , the family {Nx : x ∈ X } consisting of Nx = {x} if x ∈  and Nu i = X if 1 ≤ i ≤ n is a pointfinite open assignment for X . Now Theorem 4.3 applies.  Corollary 4.5 If X is a discrete space and α (X ) stands for the one-point compactification or the one-point Lindelöfication of X , then the space C p (α (X )) is distinguished. Note that every point-finite open assignment of X is point-finite, but not every point-finite (even clopen) cover is a point-finite open assignment (e. g., take X any infinite space and set each Nx = X ). Recall that a space X is called Eberlein compact if it is homeomorphic to a weakly compact subset of a Banach space, and Corson compact if it is homeomorphic to a compact subset of a -product of real lines. Every Eberlein compact space is Corson compact. A topological space X is called scattered if every closed non-empty subspace Y of X has an isolated point in the relative topology. We will need the following result due to Pełczy´nski and Semadeni.

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Theorem 4.6 ([45, Theorem 8.5.4]) A compact space X is scattered if and only if there is no continuous mapping of X onto [0, 1]. The following theorem has been proved in [15]. Theorem 4.7 ([15, Theorem 49]) Let X be a Corson compact space. The following assertions are equivalent. (1) X is scattered. (2) X is a scattered Eberlein compact space. (3) C p (X ) is distinguished. Proof (1) ⇒ (2). If X is a Corson scattered compact, then X is necessarily a scattered Eberlein compact space by Alster’s theorem [1, Theorem]. (2) ⇒ (3). Applying the argument of the proof of [6, Lemma 1.1] with X an arbitrary scattered Eberlein compactum, we derive that for each a ∈ X there is defined a clopen neighbourhood Va of a such that the family {Va : a ∈ X } is point-finite, with Va and Vb (clearly) distinct for distinct a, b ∈ X . This shows that {Va : a ∈ X } is a point-finite open assignment for X , so Theorem 4.3 shows that C p (X ) is distinguished. (3) ⇒ (1). Assume that C p (X ) is distinguished but X is a non-scattered Corson compact space. There is a continuous surjection f from X onto the closed interval [0, 1] (Theorem 4.6). Note that there exists a compact set Y in X which is metrizable and |Y | = c. In fact, fix any countable dense subset Q of [0, 1] and choose a countable subset P in X such that f (P) = Q. Let Y be the closure of P in X . Clearly Y is metrizable, since it is a Corson separable compact space. In addition, Y must have the cardinality of continuum since by the density of P in Y and the density of Q in [0, 1] one has f (Y ) = [0, 1]. Since C p (Y ) is not distinguished, neither is C p (X ) distinguished, by Theorem 3.4.  Recall that a family U of subsets of X is called separating if given any two distinct points x, y ∈ X , there is a member U ∈ U such that either x ∈ U and y ∈ / U , or y ∈ U and x ∈ / U. A Rosenthal’s theorem says that a compact space X is an Eberlein compact if and only if X has a σ-point-finite separating family of open Fσ -subsets. The next theorem provides a concrete class of compact spaces for which Theorem 4.3 applies. Theorem 4.8 ([15, Theorem 51]) A compact space X admits a point-finite open assignment if and only if X is a scattered Eberlein compact. Proof If X is a scattered Eberlein compact, then X admits a point-finite open assignment by the argument of Theorem 4.7 (see (2) ⇒ (3)). Assume now that X admits a point-finite open assignment ν = {Nx : x ∈ X }. Let us refine ν. Fix any u ∈ X . Then, by definition, there are only finitely many points

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xi = u in X such that u ∈ Nxi . Since X is a Tychonoff space we can choose an open Fσ -set Mu such that u ∈ Mu ⊆ Nu and Mu is disjoint from all points xi . Then the new family μ = {Mx : x ∈ X } is a point-finite open assignment for X because μ is a shrinking of ν. We show that μ separates points of X . Take different elements x, y ∈ X . If y ∈ / M y . Hence μ is a point-finite separating Mx , by our construction we have x ∈ family of open Fσ -subsets of X . Therefore, X is an Eberlein compact by Rosenthal’s characterization. Further, C p (X ) is distinguished by Theorem 4.3. Finally the space X is scattered by Theorem 4.7.  This implies the following Corollary 4.9 A non-compact locally compact space X admits a point-finite open assignment if and only if the one-point compactification of X is a scattered Eberlein compact. Denote the one-point compactification of X by α(X ). Observe that X is open in α(X ). Let ν be a point-finite open assignment of X . It is easy to see that by adding to ν the set α(X ) we obtain a point-finite open assignment of α(X ). So, the compact space α(X ) is a scattered Eberlein compact by Theorem 4.8. Conversely, it suffices to observe that if μ is a point-finite open assignment of α(X ) then ν = {U ∩ X : U ∈ μ} is a point-finite open assignment of X . Recall that a space X is strongly σ-discrete if it is the union of countably many of its closed discrete subspaces (see [48, 1.5]). Further, a topological space X is called a Q-space if every subset A of X is G δ (equivalently, Fσ ). It is very well known that every Q-set of reals is a -set. Similarly, one can prove a more general statement (for details look next Sect. 5 and [29, Proposition 4.1]): Proposition 4.10 Every Q-space is a -space. A topological space X is strongly splittable if each f ∈ R X is the pointwise limit of a sequence { f n }∞ n=1 in C (X ). Countable Tychonoff spaces and discrete spaces are strongly splittable, as well as every normal strongly σ-discrete space. Both strongly σ-discrete spaces and strongly splittable spaces are Q-spaces, hence in view of Proposition 4.10 we have Proposition 4.11 ([15, Proposition 54]) If X is a strongly splittable Tychonoff space, C p (X ) is distinguished. It turns out that a subset of a scattered Eberlein compact space consisting of G δ -points is always a Q-space ([33]). It follows from the following result. Theorem 4.12 ([33]) Assume that a topological space X admits a point-finite neighborhood assignment. If F is any subset of X consisting of points which are G δ in X , then F is a G δ -set in X .

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Proof Let {Nx : x ∈ X } be a point-finite neighborhood assignment in X . For every point x ∈ F fix the sequence of open sets {Un (x) : n ∈ ω} such that U0 (x) ⊆ Nx , Un+1 (x) ⊆ Un (x) and

{Un (x) : n ∈ ω} = {x}.



Define open sets Vn = x∈F Un (x) for every n ∈ ω. We claim that n∈ω Vn = F. Indeed, let y ∈ / F be any point. There are at most finitely many points x1 , x2 , . . . , xk in F such that y ∈ Nxi . Since n∈ω Un (xi ) = {xi } for each i = 1, 2, . . . , k, there is / Un (xi ) for every n > n 0 . Finally, y ∈ / Vn one index n 0 large enough such that y ∈ for every n > n 0 .  Note that Theorem 4.12 generalizes [31, Theorem 1].

5 C p (X) is Distinguished if and Only if X is a -Space In this section we characterize distinguished spaces C p (X ) in terms of the space X . Most of presented here results were already published in papers [28, 29]. Another characterization was presented by Ferrando and Saxon [16, Theorem 5]. We start with the following concept reintroduced in [28]. Definition 5.1 ([28]) A topological space X is said to be a -space if for every decreasing sequence {Dn : n ∈ ω} of subsets of X with empty intersection, there is a decreasing sequence {Vn : n ∈ ω} consisting of open subsets of X , also with empty intersection, and such that Dn ⊂ Vn for every n ∈ ω. The class of all -spaces is denoted by . The original definition of a -set of the real line R is due to G. M. Reed and E. K. van Douwen (see [40]). Recall also here that a Q-set X is a subset of reals such that each subset of X is G δ . A subset X of reals is called a λ-set if each countable subset of X is G δ in X . It is well known that every -set is a λ-set. It should be pointed out that K. Kuratowski constructed uncountable λ-sets in ZFC. On the other hand, F. Hausdorff (1933) showed that the cardinality of an uncountable Q-set X has to be strictly smaller than the continuum, so assuming the Continuum Hypothesis (CH) there are no uncountable Q-sets. Martin’s Axiom plus the negation of the Continuum Hypothesis implies that every subset X ⊂ R of cardinality less than continuum is a Q-set. No -set X can have cardinality continuum [39]. Consequently, under Martin’s Axiom, every subset of reals that is a -set is also a Q-set. -sets of reals have been used and investigated thoroughly in the study of two the most basic and central constructions in general topology: the Moore–Nemytskii plane and the Pixley-Roy topology. For example, if M(X ) is the subspace of the Moore–Nemytskii plane, which is obtained by using only a subset X ⊂ R of the

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x-axis, G. M. Reed observed that M(X ) is countably paracompact if and only if X is a -set [40]. More details about Q-sets and -sets can be found in [19]. Of course, there are plenty of nonmetrizable -spaces with non-G δ subsets, in ZFC [28]. Recently J. K¸akol and A. Leiderman have shown that the notion of -spaces plays a key role in the study of distinguished C p -spaces, [28]. First recall the following useful concept. A collection of sets {Uγ : γ ∈ } is called an expansion of a collection of sets {X γ : γ ∈ } in X if X γ ⊆ Uγ ⊆ X for every index γ ∈ . The following theorem characterizing -spaces X as those for which the corresponding space C p (X ) is distinguished plays a crucial role in the sequel. Independently and simultaneously an analogous description of distinguished C p -spaces (but formulated in different terms) appeared in [16, Theorem 5]. Theorem 5.2 ([28, Theorem 2.1]) For a Tychonoff space X , the following conditions are equivalent: (1) C p (X ) is distinguished. (2) Any countable partition of X admits a point-finite open expansion in X . (3) Any countable disjoint collection of subsets of X admits a point-finite open expansion in X . (4) X is a -space. Proof Observe that every collection of pairwise disjoint subsets of  X , {X γ : γ ∈ } can be extended to a partition by adding a single set X ∗ = X \ {X γ : γ ∈ }. If the obtained partition admits a point-finite open expansion in X , then removing one open set we get a point-finite open expansion of the original disjoint collection. This shows evidently the equivalence (2) ⇔ (3). Assume now that (3) holds. Let {Dn : n ∈ ω} be a decreasing sequence subsets of X with empty intersection. Define X n = Dn \ Dn+1 for each n ∈ ω. By assumption, a disjoint collection {X n : n ∈ ω} admits a point-finite open expansion {Un : n ∈ ω} in X . Then {Vn = {Ui : i ≥ n} : n ∈ ω} is an open decreasing expansion in X with empty intersection. This proves the implication (3) ⇒ (4). Next we show (4) ⇒ (2).  Let {X n : n ∈ ω} be any countable partition of X . Define D0 = X and Dn = X \ {X i : i < n}. Then X n ⊂ Dn for every n, the sequence {Dn : n ∈ ω} is decreasing and its intersection is empty. Assuming (4), we find an open decreasing expansion {Un : n ∈ ω} of {Dn : n ∈ ω} in X such that {Un : n ∈ ω} = ∅. For every x ∈ X there is n such that x ∈ / Um for each m > n, it means that {Un : n ∈ ω} is a point-finite expansion of {X n : n ∈ ω} in X . This finishes the proof (3) ⇒ (4) ⇒ (2) ⇔ (3). Now we prove the implication (1) ⇒ (2). Let {X n : n ∈ ω} be any countable partition of X . Fix any function f ∈ R X which satisfies the following conditions: for each n ∈ ω and every x ∈ X n the value of f (x) is greater than n. By assumption, there is a bounded subset B of C p (X ) such that f ∈ clR X (B). Hence, for every n ∈ ω and every point x ∈ X n , there exists f x ∈ B such that f x (x) > n. But f x is a continuous function, therefore there is an open neighbourhood Ux ⊂ X of x

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such that  f x (y) > n for every y ∈ Ux . We define an open set Un ⊂ X as follows: Un = {Ux : x ∈ X n }. Evidently, X n ⊆ Un for each n ∈ ω. If we assume that the open expansion {Un : n ∈ ω} is not point-finite, then there exists a point y ∈ X such that there are infinitely many numbers n with y ∈ Uxn for some xn ∈ X n . This means that sup{g(y) : g ∈ B} = ∞, which contradicts the boundedness of B. It remains to prove (2) ⇒ (1). By Theorem 3.1, we need to show that for every mapping f ∈ R X there is a bounded set B ⊂ C p (X ) such that f ∈ clR X (B). If there exists a constant r > 0 such that sup{| f (x)| : x ∈ X } < r , then we take B = {h ∈ C(X ) : sup{|h(x)| : x ∈ X } < r }. It is easy to see that B is as required. Let f ∈ R X be unbounded. Denote by Y0 = ∅ and Yn = {x ∈ X : n − 1 ≤ | f (x)| < n} for each non-zero n ∈ ω. Define ϕ : X → ω by the rule: if Yn = ∅ then ϕ(x) = n for every x ∈ Yn . So, | f | < ϕ. Put X n = ϕ−1 (n) for each n ∈ ω. Note that some sets X n might happen to be empty, but the collection {X n : n ∈ ω} is a partition of X with countably many nonempty X n -s. By our assumption, there exists a point-finite open expansion {Un : n ∈ ω} of the partition {X n : n ∈ ω}. Define F : X → ω by F(x) = max{n : x ∈ Un }. Obviously, f < F. Finally, we define B = {h ∈ C p (X ) : |h| ≤ F}. Then f ∈ clR X (B), because for every finite subset K ⊂ X there is a function h ∈ B such that f  K = h  K . Indeed, given a finite subset K ⊂ X , let {Vx : x ∈ K } be the family of pairwise disjoint open sets such that x ∈ Vx ⊂ Uϕ(x) for every x ∈ K . For each x ∈ K , fix a continuous function h x : X → [−ϕ(x), ϕ(x)] such that h x (x) = f (x) and h x is equal to the constant value 0 on the closed set X \ Vx . One can verify that h = x∈K h x ∈ B is as required.  The rest of the paper deals with the study of -spaces, which apparently provides a nice extension of the research around -sets, and is strictly connected with the study of distinguished spaces C p (X ) via Theorem 5.2. The following consequence of Theorem 5.2 provides another approach to Theorem 3.4. Corollary 5.3 Let Z be any subspace of X . If X belongs to the class , then Z also belongs to the class . Proof If {Z γ : γ ∈ } is any collection of pairwise disjoint subsets of Z and there exists a point-finite open expansion {Uγ : γ ∈ } in X , then obviously {Uγ ∩ Z : γ ∈ } is a point-finite expansion consisting of the sets relatively open in Z . It remains to apply Theorem 5.2.  Remark 5.4 Our Theorem 5.2 implies one of the results obtained in Theorem 4.3 stating that if X admits a point-finite open assignment {Nx : x ∈ X } then C p (X ) is distinguished. Indeed, let {X γ : γ ∈ } be any collection of pairwise disjoint subsets of X .  Define Uγ = {Nx : x ∈ X γ }. It is easily seen that {Uγ : γ ∈ } is a point-finite open expansion in X . Applying Theorem 5.2, we conclude that C p (X ) is distinguished. In order to present next results about -spaces we need recall some known concepts and facts. A continuous surjection π : X → Y is called irreducible (see [45,

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Definition 7.1.11]) if for every closed subset F of X the condition π(F) = Y implies F = X. Proposition 5.5 ([45, Proposition 7.1.13]) Let X be a compact space and let π : X → Y be a continuous surjection. Then there exists a closed subset F of X such that π(F) = Y and the restriction π|F : F → Y is irreducible. Proposition 5.6 ([45, Proposition 25.2.1]) Let X be a compact space and let π : X → Y be a continuous surjection. Then π is irreducible if and only if whenever E ⊂ X and π(E) is dense in Y , then E is dense in X . ˇ Recall that a Tychonoff space X is Cech-complete if X is a G δ -set in some (equivalently, any) compactification of X , (see [10, 3.9.1]). It is well known that every ˇ locally compact space and every completely metrizable space is Cech-complete. ˇ Theorem 5.7 ([28, Theorem 3.4]) Every Cech-complete (in particular, compact) -space is scattered. Proof Assume X is compact. On the contrary, assume that X is not scattered. By Theorem 4.6, there is a continuous mapping π from X onto the segment [0, 1]. Then, by Proposition 5.5, there exists a closed subset F of X such that π(F) = [0, 1] and the restriction π|F : F → [0, 1] is irreducible. Since X ∈  the compact space F also belongs to , by Corollary 5.3. Without loss of generality we may assume that F is X itself and π : X → [0, 1] is irreducible.  Let {X n : n ∈ ω} be a partition of [0, 1] into dense sets. Put Yn = k≥n X k , and Z n = π −1 (Yn ) for

all n ∈ ω. Then all sets Z n are dense in X by Proposition 5.6 and the intersection n∈ω Z n is empty. Every compact space X is a Baire space, i.e. the Baire category theorem holds in X . Hence

if {Un : n ∈ ω} is any open expansion of {Z n : n ∈ ω}, then the intersection n∈ω Un is dense in X . In view of our Theorem 5.2 this conclusion contradicts the assumption X ∈ , and the proof follows. ˇ Assume X is any Cech-complete space. By the first step we deduce that every ˇ compact subset of X is scattered. On the other hand, each Cech-complete space X is scattered if and only if every compact subset of X is scattered.  In [29] we asked whether every pseudocompact -space is scattered. Leiderman and Tkachuk [34] proved the following partial answer to the question. Theorem 5.8 ([34]) If X is a pseudocompact -space, then every countable subset of X is scattered. This yields the following Corollary 5.9 ([34]) Every pseudocompact space with countable tightness which is a -space must be scattered. Below we show that there exist scattered compact spaces X ∈  which are not Eberlein compact, and there exist compact scattered spaces X ∈ / . Our first example will be the one-point compactification of an Isbell–Mrówka space (A).

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We recall briefly here the construction and basic properties of (A). Let A be an almost disjoint family of subsets of the set of natural numbers N and let (A) be the set N ∪ A equipped with the topology defined as follows. For each n ∈ N, the singleton {n} is open, and for each A ∈ A, a base of neighbourhoods of A is the collection of all sets of the form {A} ∪ B, where B ⊂ A and |A \ B| < ω. The space (A) is then a first-countable separable locally compact Tychonoff space. If A is a maximal almost disjoint (MAD) family, then the corresponding Isbell–Mrówka space (A) would be in addition pseudocompact. Theorem 5.10 ([28, Theorem 3.10]) Let A be any uncountable almost disjoint (in particular, MAD) family of subsets of N and let X be the one-point compactification of the corresponding first-countable separable locally compact Isbell–Mrówka space (A). Then X ∈  and X is not an Eberlein compact space. Proof It can be easily verified that X can be represented as a countable union of scattered Eberlein compact spaces and make use of Theorem 4.7 and Proposition 5.18. The above space X is not an Eberlein compact space, since every separable Eberlein compact space is metrizable, while (A) is metrizable if and only if A is countable.  There are examples of scattered compact spaces which are not in the class . Theorem 5.11 ([28, Theorem 3.12]) The space [0, ω1 ) is not in the class . Consequently, [0, ω1 ] is not in the class . There is even a scattered compact space X such that its scatteredness height is finite, but X ∈ /  [33]. Next we analyse when metrizable spaces belong to class . First recall that a topological space X is said to be σ -scattered if X can be represented as a countable union of scattered subspaces and X is called strongly σ -discrete if it is a union of countably many of its closed discrete subspaces. Strongly σ-discreteness of X implies that X is σ-scattered. For metrizable X , these two properties are equivalent, see [46]. This implies the following result Proposition 5.12 ([28, Proposition 4.1]) Any σ-scattered metrizable space X belongs to the class . Remark 5.13 The existence of an uncountable separable metrizable -space is equivalent to the existence of an uncountable -set. Proof Note that every separable metrizable space homeomorphically embeds into a Polish space Rω and the latter space is a one-to-one continuous image of the set of irrationals J . Therefore, if M is an uncountable separable metrizable space, then there exist an uncountable set X ⊂ R and a one-to-one continuous mapping from X onto M. It is easy to see that X is a -set provided M is a -space. 

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In [28] we investigated whether the class  is invariant under the basic topological operations, including continuous images, closed continuous images, countable unions and finite products. We recall some of results concerning this subject; most of them have been proved in [28, 29]. We start first with the following result from [23]. Proposition 5.14 ([23]) There exists in ZFC a MAD family A on N such that the corresponding Isbell–Mrówka space (A) admits a continuous mapping onto the closed interval [0, 1]. This shows that the class  is not invariant under continuous images even for first-countable separable locally compact pseudocompact spaces. Proposition 5.15 ([29, Example 3.17]) There exists an Isbell–Mrówka space Z which is almost compact in the sense that the one-point compactification of Z coincides with β Z . In [29] we proved the following general theorem providing several applications. Theorem 5.16 ([29, Theorem 2.1]) Let X be any -space and ϕ : X → Y be a continuous surjection such that ϕ(F) is an Fσ -set in Y for every closed set F ⊂ X ; then Y is also a -space. This immediately yields the following consequence. Corollary 5.17 Any continuous image of a compact -space is also a -space. We can provide even a stronger result as mentioned in Corollary 5.17. This will be a consequence of the following Proposition 5.18 ([29]) Assume that X is a countable union of closed subsets X n , where each X n belongs to the class . Then X also belongs to . In particular, a countable union of compact -spaces is also a -space. Proof Denote by Z the free topological union of the spaces X n , n ∈ ω. Note that that Z ∈ , by Theorem 5.2. The space Z admits a natural continuous mapping ϕ onto X . Since ϕ(F) is an Fσ -set in X for every closed set F ⊂ Z , we apply Theorem 5.16 to deduce that X ∈ .  Corollary 5.19 Let X be a σ-compact -space and Y be a continuous image of X . Then Y also is a -space. Corollary 5.20 A σ-product of any family consisting of scattered Eberlein compact spaces is a -space. Proof A σ-product is a countable union of σn -products, where σn -product includes elements of the product which support consists of at most n points, n ∈ ω. Every σn product of scattered Eberlein compact spaces is again a scattered Eberlein compact, therefore it is a -space. It remains to apply Proposition 5.18. 

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Corollary 5.21 Let X be a Lindelöf subspace of a σ-product of any family consisting of scattered Eberlein compact spaces and Y be a continuous image of X . Then Y also is a -space. Proof The space X is equal to the countable union of its closed subspaces X n , where X n is the intersection of X with σn -product. Every Lindelöf subspace of a scattered Eberlein compact is necessarily σ-compact, by a recent result of Tkachuk [49]. Finally, Y is a -space by applying Corollary 5.19.  ˇ Corollary 5.22 Let X be a Lindelöf Cech-complete -space and Y be a continuous image of X . Then Y also is a -space. ˇ Proof We know already that each Cech-complete -space is scattered. Now we use ˇ the well-known fact that every Lindelöf Cech-complete scattered space is σ-compact, see [3, Theorem 4.5]. To complete the proof it is enough to apply Corollary 5.19.  Corollary 5.23 Let Z be the product of a -space X with a σ-closed discrete space (in particular, a countable space) Y . Then Z also is a -space.  Proof Let Y = n∈ω Yn , where each Yn is a closed and discrete subset of Y . Denote by Z n = X × Yn . It is clear that each Z n is closed in Z and Z n ∈ . We get that Z  is a countable union of closed -spaces Z n , so Proposition 5.18 applies. Unfortunately, we do not know whether the class  is invariant under finite products. Problem 5.24 ([28]) Is the class  invariant under finite products? A topological space X is called ω-bounded if the closure of every countable subset of X is compact. Another application of Theorem 5.16 is the following theorem. Theorem 5.25 ([29, Theorem 4.1]) Every ω-bounded -space is compact. Proof Assume that the claim fails. Then, by a result of Burke and Gruenhage [20, Lemma 1], X contains a subset Z which is a perfect preimage of the ordinal space [0, ω1 ). We conclude that a -space Z can be mapped by a continuous closed mapping onto [0, ω1 ). By Theorem 5.16 this would mean that [0, ω1 ) ∈ , a contradiction  since [0, ω1 ) is not in  by Theorem 5.11. A very similar argument as presented in Theorem 5.25 can be applied to get the following theorem motivated by Corollary 5.9. Theorem 5.26 ([29, Theorem 4.2]) Every compact -space has countable tightness. Proof A compact space has countable tightness if and only if it does not contain a perfect preimage of [0, ω1 ) (see [4]). To complete the proof see the argument presented in the proof of Theorem 5.25. 

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A very natural question arises whether Theorem 5.26 can be generalized for countably compact spaces. A positive answer follows from the Proper Forcing Axiom (PFA), due to the results of Balogh [4] (see also [5]). Theorem 5.27 ([29, Theorem 4.3]) (PFA) (1) Every countably compact -space is compact. (2) Every countably compact -space has countable tightness. (3) Every countably compact -space (hence, every compact -space) is sequential. The following interesting Theorem 5.28 does not require extra set-theoretic assumptions. Theorem 5.28 ([29]) Every countably compact -space is scattered. Proof Assume that a countably compact space X is not scattered. Recall that every countably compact space X is pseudocompact. Hence there exists a closed subset K ⊂ X and a continuous surjective mapping ϕ from K onto [0, 1], by applying [32, Proposition 5.5]. Since every closed subset F of K is a countably compact space, its continuous image ϕ(F) is a countably compact subset of [0, 1], therefore ϕ(F) is compact. We conclude that the restriction ϕ|K is a closed continuous mapping from K onto [0, 1]. This evidently contradicts Theorem 5.16, since [0, 1] ∈ / .  This can be used to show the following example which is also motivated by Theorem 5.8. Example 5.29 The space βω \ ω contains a dense countably compact not scattered subspace K which is not a -space and every countable subset of K is scattered. Indeed, by [26, Theorem 1.1, Examples] the space βω \ ω contains a dense countably compact space K which is not scattered although all countable subsets of K are scattered. Now it is enough to apply Theorem 5.28. On the other hand, we note the following result characterizing first-countable compact -spaces. Proposition 5.30 ([28]) If X is a first-countable compact space, then X ∈  if and only if X is countable. Proof If X ∈ , then X is scattered, by Theorem 5.7. By [45, Theorem 8.6.10], a first-countable compact space is scattered if and only if it is countable. This proves one direction of the proof. The converse is known [14] and follows from the fact that any countable space X = {xn : n ∈ ω} admits a point-finite open expansion. Indeed, define X n = {xi : i ≥ n}. Then the family {X n : n ∈ ω} is a point-finite open expansion of X . Now it suffices to mention Remark 5.4. 

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6 More Examples of Non -Spaces This section presents some results due to A. Leiderman and P. Szeptycki presented in [33] which extend also several obtained results from [15] concerning example spaces X for which C p (X ) not being distinguished. We know already that one-point compactification of any Isbell-Mrowka space is a separable -space. It follows that separable compact -spaces with |X | = c do exist in ZFC. Nevertheless, no -set of reals can have cardinality c, see [39]. This latter fact can be extended for a larger class of topological spaces as follows. / , where o(X ) means the cardinality Theorem 6.1 ([33]) If o(X )ℵ0 ≤ |X |, then X ∈ of the family of all open sets in X . Proof The idea of the proof uses some argument presented in [39]. Denote |o(X )ℵ0 | = λ. Let X = {xα : α < τ } and enumerate by {{Unα : n ∈ ω} : α < λ} all countable sequences of open subsets of X with empty intersection. Assume that X ∈ . We know by assumption that λ ≤ τ . For every α < λ choose an n(α) ∈ ω such that α / Un(α) . xα ∈

Now, define An = {xα : n(α) ≥ n}. Clearly, n∈ω An = ∅. If there existed an α , α < λ such that An ⊂ Unα , for each n ∈ ω, then we would have xα ∈ An(α) ⊂ Un(α) a contradiction.  Next Proposition 6.2 (being an application of Theorem 6.1) extends some results from [15], for example Corollary 30 and Proposition 37 (from [15]). Proposition 6.2 ([33]) (a) Let X be a hereditarily separable space. If |X | = c, then X∈ / . (b) Let X be a separable hereditarily Lindelöf space. If |X | = c, then X ∈ / . Proof (a) Note that o(X ) ≤ |X |hd(X ) holds for each X , [22]. Since hd(X ) = ℵ0 we get that o(X )ℵ0 ≤ cℵ0 ×ℵ0 = |X |, and Theorem 6.1 applies. (b) The inequalities w(X ) ≤ 2d(X ) and o(X ) ≤ w(X )h L(X ) always hold for any (reg ular) X , see [22]. Hence o(X )ℵ0 ≤ cℵ0 ×ℵ0 = |X |, and Theorem 6.1 applies. Let S denote the Sorgenfrey line. It has been shown in [15, Example 34] that S∈ / . Note however that the proof presented in [15] is valid for any subset of S containing a segment but although fails for more complicated subspaces of S. By Proposition 6.2(a) we have an improvement of [15, Example 34]. Corollary 6.3 Let X be any subspace of the Sorgenfrey line S with |X | = c. Then X∈ / . Note another application of Theorem 6.1. Recall that nw(X ) denotes the network weight of X . We have that always nw(X ) ≥ ℵ0 holds because X is assumed to be infinite. Next Corollary 6.4 extends [15, Corollary 31]. Corollary 6.4 Let X be any space. If 2nw(X ) ≤ |X | then X ∈ / .

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Proof Fix a network N in X with |N | = nw(X  ). If U ⊂ X is any open set then there is a subfamily NU ⊂ N such that U = NU . This means that always o(X ) ≤ 2nw(X ) . It follows that o(X )ℵ0 ≤ 2nw(X )×ℵ0 = 2nw(X ) . Hence, assuming 2nw(X ) ≤ |X | we obtain that o(X )ℵ0 ≤ |X | and then Theorem 6.1 applies.  Recall that a space is ω-resolvable if it has no isolated points and can be partitioned into countably many dense subsets. Proposition 6.5 If X is Baire and ω-resolvable, then X ∈ / . Proof Fix a partition of X into countably many dense sets Dn . If Un ⊇ Dn then Un is dense open and so n Un = ∅ and so {Dn : n ∈ ω} has no point-finite open expansion.  Note also that Filatova proved [18] that a Lindelöf regular spaces with all open sets uncountable is resolvable (i.e., can be partitioned into two dense subsets) and this was improved by Juhasz et al [27] to ω-resolvable. Since Baire spaces without isolated points have all open subsets uncountable, we obtain the following extension of the fact that compact spaces in class  are scattered. Indeed, we have Corollary 6.6 Lindelöf Baire spaces without isolated points are not -spaces.

7 -spaces vs properties of spaces C p (X) This section deals with the following question posed and studied in the paper [29]: Which topological properties related to being a  -space are preserved by the relation of l-dominance? Following Arkhangell’ski [2] we say that a Tychonoff space is l-dominated by a Tychonoff space X if there exits a continuous linear surjection from C p (X ) onto C p (Y ). The main result of this section is the following. Theorem 7.1 ([29, Theorem 3.1]) Assume that Y is l-dominated by X . If X is a -space, then Y also is a -space. The proof of Theorem 7.1 uses the following simple Lemma 7.2 below. Lemma 7.2 Let X and Y be two sets and let E ⊂ R X and F ⊂ RY be dense vector subspaces of R X and RY , respectively. Assume that T : E −→ F is a continuous linear surjection between lcs E and F. Then T admits a continuous linear surjective  : R X −→ RY . (unique) extension T Proof Note the following well-known properties of R X which will be used. Property 1. Every closed vector subspace H of R X is complemented in R X and the quotient R X /H is linearly homeomorphic to the product R Z for some set Z , see [38, Corollary 2.6.5, Theorem 2.6.4].

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Property 2. The product topology on R X is minimal, i.e. R X does not admit a weaker Hausdorff locally convex topology, see [38, Corollary 2.6.5(i)]. Property 3. The space RY fulfills the extension property, i.e. if M is a vector subspace of a lcs L, then every continuous linear mapping T : M −→ RY admits a continuous  : L −→ RY , see [36, Theorem 10.1.2 (a)]. linear extension T  : R X −→ RY of T By Property 3, there exists a continuous linear extension T (R X ). such that F ⊂ T  is a surjective mapping. Next, denote by ϕ : R X / ker(T ) −→ RY We prove that T ), the injective mapping associated with the quotient mapping Q : R X −→ R X / ker(T ) is the kernel of T  and ϕ ◦ Q = T . where ker(T By Property 1, the space R X / ker(T ) is linearly homeomorphic to the product R Z for some set Z . Hence we may assume that ϕ is a continuous linear bijection from (R X ) of RY . This implies that on T (R X ) there exists a R Z onto a dense subspace T X (R ), ξ) is linearly homeomorphic stronger locally convex topology ξ such that (T with R Z . However, by Property 2, the space R Z does not admit a weaker Hausdorff (R X ) is isomorphic to the complete lcs R Z . Finally, locally convex topology, hence T X Y  is a surjection. (R ) is closed in R and then T  T Proof of Theorem 7.1 Let T : C p (X ) −→ C p (Y ) be a continuous linear surjection.  : R X −→ RY . Denote the extension of T which is provided by Lemma 7.2 by T Now Theorem 3.1 applies to show that Y is a -space. (g) = f . There Indeed, take arbitrary f ∈ RY . Then, there exists g ∈ R X with T exists a bounded set B ⊂ C p (X ) such that g ∈ B to see that A is bounded and f ∈ A

RY

RX

. We define A = T (B). It is easy

which means that C p (Y ) is distinguished. 

Remark 7.3 In Theorem 7.1 the linearity of T : C p (X ) → C p (Y ) cannot be dropped. Indeed, let Y be a metrizable separable space with |Y | = c and such that Ck (Y ) is analytic, i.e. Ck (Y ) is a continuous image of the irrationals J . / . The space C p (S) Let S be the space {0} ∪ {n −1 : n ∈ N}. Then S ∈  but Y ∈ contains a closed homeomorphic copy of the space J . Since Ck (Y ) is analytic, there is a continuous surjection L : J → Ck (Y ) which extends to a continuous surjection T : C p (S) → Ck (Y ), by applying the classic Dugundji’s theorem. Theorem 7.4 ([29]) Let X and Y be normal spaces and assume that Y is l-dominated by X . If X is a Q-space, then Y also is a Q-space. Proof A normal space X is a Q-space if and only X is strongly splittable, i.e. for every f ∈ R X there exists a sequence S = { f n : n ∈ ω} ⊂ C p (X ) such that f n → f in R X , by [48, Problems 445, 447]. Let T : C p (X ) −→ C p (Y ) be a continuous linear surjection. Denote the extension  : R X −→ RY . of T which is supplied by Lemma 7.2 by T

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(g) = f . Then there exists a Take any f ∈ RY . There exists g ∈ R X with T sequence B ⊂ C p (X ) converging to g in R X . It is easy to see that the sequence  A = T (B) ⊂ C p (Y ) converges to f in RY . Corollary 7.5 ([29]) Let X and Y be metrizable spaces (in particular, subsets of R) and assume that Y is l-dominated by X . If X is a Q-space, then Y also is a Q-space. Our next statement is a combination of a few known results, while we apply Theorem 7.1 in order to obtain the scatteredness of a target space. Proposition 7.6 ([29]) Assume that Y is l-dominated by X . If X is a scattered Eberlein compact, then Y also is a scattered Eberlein compact. Proposition 7.7 ([29]) Let X and Y be metrizable spaces and assume that Y is l-dominated by X . If X is scattered, then Y also is scattered. We complete this survey paper with the following two questions posed in [28]. Here we consider the Banach spaces C(X ). Problem 7.8 Characterize compact -spaces in terms of a suitable property of the Banach space C(X ) or its dual. Recall that if X is a compact -space, then X is scattered. But then C(X ) is an Asplund space, see [11, Theorem 12.29]. Problem 7.9 Assume that X and Y are compact spaces and T : C(X ) → C(Y ) is a continuous linear surjection. Assume that X is a -space. Is Y a -space? Note that if X is a -space, then X is scattered and then Y is scattered, too. Indeed, the property for Banach spaces of being an Asplund space is inherited by Hausdorff quotients and isomorphic copies, see [35]. Moreover, if the answer of the above problem is negative, i.e. Y is not a -space, then by Theorem 7.1 there is no continuous linear map from C p (X ) onto C p (Y ).

References 1. Alster, K.: Some remarks on Eberlein compacts. Fund. Math. 104, 43–46 (1979) 2. Arkhangel’skii, A.V.: C p -theory. In: Hušek, M., van Mill, J. (eds.) Recent progress in general topology, pp. 1–56. Elsevier, Amsterdam (1992) 3. Aviles, A., Guerrero Sanchez, D.: Are Eberlein-Grothendieck scattered spaces σ-discrete? Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 108, 849–859 (2014). https://doi.org/ 10.1007/s13398-013-0146-2 4. Balogh, Z.: On compact Hausdorff spaces of countable tightness. Proc. Am. Math. Soc. 105, 755–764 (1989) 5. Balogh, Z., Dow, A., Fremlin, D.H., Nyikos, P.J.: Countable tightness and proper forcing. Bull. Am. Math. Soc. (New Series) 19, 295–298 (1988) 6. Bell, M., Marciszewski, W.: On scattered Eberlein compact spaces. Israel J. Math. 158, 217–224 (2007)

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Variational Convexity of Functions in Banach Spaces Pham Duy Khanh, Vu Vinh Huy Khoa, Boris S. Mordukhovich, and Vo Thanh Phat

Abstract This paper addresses the study and characterizations of variational convexity of extended-real-valued functions on Banach spaces. This notion has been recently introduced by Rockafellar, and its importance has been already realized and applied to continuous optimization problems in finite-dimensional spaces. Variational convexity in infinite-dimensional spaces, which is studied here for the first time, is significantly more involved and requires the usage of powerful tools of geometric functional analysis together with variational analysis and generalized differentiation in Banach spaces. Keywords Functional analysis and continuous optimization · Variational analysis and generalized differentiation · Extended-real-valued functions · Variational convexity · Monotone operators · Moreau envelopes Mathematics Subject Classification (2000) 49J52 · 49J53 · 46B20 · 46B10 · 46A55

Vu Vinh Huy Khoa and Vo Thanh Phat—Research of this author was partly supported by the US National Science Foundation under grants DMS-1808978 and DMS-2204519. Boris S. Mordukhovich—Research of this author was partly supported by the US National Science Foundation under grants DMS-1808978 and DMS-2204519, and by the Australian Research Council under Discovery Project DP-190100555. P. D. Khanh Member of the Group of Analysis and Applied Mathematics, Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam e-mail: [email protected]; [email protected] V. V. H. Khoa · B. S. Mordukhovich (B) · V. T. Phat Department of Mathematics, Wayne State University, Detroit, Michigan, USA e-mail: [email protected] V. V. H. Khoa e-mail: [email protected] V. T. Phat e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_11

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1 Introduction The notion of variational convexity for extended-real-valued functions has been recently introduced and studied by Rockafellar [22] in finite-dimensional spaces. This notion is different from the standard local convexity of functions while offering instead a new insight on local behavior of the function in question via a certain local maximal monotonicity of its subdifferential. Such a novel viewpoint occurs to be useful in applications to various issues in continuous optimization including the design and justification of numerical algorithms; see [23, 24]. Useful subdifferential characterization of variational convexity are given in [22]. More recently, it has been revealed in [13] that the variational convexity of extended-real-valued function in finite-dimensional spaces is equivalent to the standard local convexity of their Moreau envelopes. Furthermore, paper [13] contains applications of the obtained characterization of variational convexity via Moreau envelopes to variational sufficiency in constrained nonsmooth optimization. We are not familiar with any publications on variational convexity of functions defined on infinite-dimensional spaces, which is the main topic of this paper. The main goals here are to establish such characterizations in general frameworks of Banach spaces and extend in this way finite-dimensional characterizations from both papers [13, 22]. As seen below, accomplishing these goals is a highly challenging task, which requires the usage and developments of delicate advanced techniques of Banach space geometry and variational analysis in infinite dimensions with considering a variety of appropriate settings in Banach spaces. The subsequent material is organized as follows. Section 2 collects the basic definitions, preliminaries, and discussions from functional and variational analysis, which are broadly used in the paper. In Sect. 3, we define and discuss the basic notion of variational convexity of extended-real-valued functions on Banach spaces and establish its graphical subdifferential characterization in the case of reflexive spaces with the necessity part holding in general Banach spaces. Section 4 provides characterizations of variational convexity via local maximal monotonicity of subdifferential and local convexity of Moreau envelopes of functions defined on uniformly convex spaces, where some implications hold in more general Banach space settings. Throughout of the paper, we use the standard notation and terminology of variational analysis and Banach space theory; see, e.g., [6, 16, 20, 25].

2 Preliminaries and Initial Discussions Unless otherwise stated, the space X in question is Banach with the norm  ·  and its topological dual X ∗ , where ·, · indicates the canonical pairing between w X and X ∗ . The symbol → refers to the strong/norm convergence, while → and w∗

→ signify the weak and weak∗ convergence, respectively. Given a set-valued

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mapping/multifunction F : X ⇒ X ∗ from a Banach space X to it dual space X ∗ , the notation    w∗ Lim sup F(x) := x ∗ ∈ X ∗  ∃ seqs. xk → x, xk∗ → x ∗ with xk∗ ∈ F(xk ), k ∈ IN

(1)

x→x

signifies the (sequential) Painlevé-Kuratowski outer limit of F(x) as x → x. Note that the symbol := means “equal by definition” and that N := {1, 2, . . .} denotes the set of all natural numbers. Consider further the duality mapping J : X ⇒ X ∗ between X and X ∗ defined by    J (x) := x ∗ ∈ X ∗  x ∗ , x = x2 = x ∗ 2 , x ∈ X,

(2)

which plays a significant role in what follows. The notation Br (x) and Br (x), stand, respectively, for the closed ball and open ball centered at x with radius r > 0. Given an extended-real-valued function ϕ : X → IR := (−∞, ∞], we always assume that ϕ is proper, i.e., dom ϕ := {x ∈ X | ϕ(x) < ∞} = ∅. For any ε ≥ 0, the ε-subdifferential of ϕ at x¯ ∈ dom ϕ is defined by    ∗ ε ϕ(x) := x ∗ ∈ X ∗  lim inf ϕ(x) − ϕ(x) − x , x − x ≥ −ε . ∂ x→x x − x

(3)

When ε = 0 in (3), this construction is called the (Fréchet) regular subdifferential of  ϕ at x and is denoted by ∂ϕ(x). The (Mordukhovich) limiting/basic subdifferential of ϕ at x is defined via the sequential outer limit (1) by ε ϕ(x), ∂ϕ(x) ¯ := Lim sup ∂ ϕ

(4)

x →x¯ ε↓0 ϕ  where x → x¯ means that x → x¯ with ϕ(x) → ϕ(x). ¯ In the case where ∂ϕ(x) = ∂ϕ(x), the function ϕ is called lower regular at x; see [16]. It is well known that both (3) when ε = 0 and (4) reduce to the classical gradient ∇ϕ(x) for C 1 -smooth, i.e., continuously differentiable (in fact strictly differentiable) functions. If ϕ is convex, then both of these subdifferentials reduce to the standard subdifferential of convex analysis defined as

   ∂ϕ(x) := x ∗ ∈ X ∗  x ∗ , x − x ≤ ϕ(x) − ϕ(x) for all x ∈ X . Thus the aforementioned two classes of functions exhibit lower/subdifferential regularity, while the latter collection of extended-real-valued functions is much broader; see, e.g., [16, 17, 25]. It follows from [16, Theorem 2.34] that we can equivalently put ε = 0 in (4) if ϕ is lower semicontinuous (l.s.c.) around x and the space X is Asplund, i.e., a Banach space where each separable subspace has a separable dual. This class

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of spaces is fairly large including, e.g., every reflexive Banach space and every Banach space which dual is separable; see [5, 10, 16, 20] with more details and the references therein. It has been well recognized in variational analysis that the limiting subdifferential (4) and the associated constructions for sets and set-valued mappings enjoy full calculi in Asplund spaces with a variety of applications presented in the two-volume book [16], while finite-dimensional specifications can be found in [17, 25]. Some useful results for (4) hold in general Banach spaces; see, e.g., [16, Chap. 1]. The next material is mostly taken from [25] in the case of finite dimensions and from [3, 4] where extensions of these notions are studied in infinite-dimensional spaces. An l.s.c. function ϕ : X → IR on a Banach space X is prox-bounded if it majorizes a quadratic function, i.e., ϕ(x) ≥ αx − x2 + β for some α, β ∈ IR, and x ∈ X.

(5)

An l.s.c. function ϕ is prox-regular at x¯ ∈ dom ϕ for x¯ ∗ ∈ ∂ϕ(x) if there exist numbers ε > 0 and r ≥ 0 such that we have the estimate r ϕ(x) ≥ ϕ(u) + u ∗ , x − u − x − u2 2

(6)

¯ and (u, u ∗ ) ∈ gph ∂ϕ ∩ (Bε (x) × Bε (x ∗ )) with ϕ(u) < ϕ(x) ¯ + ε. for all x ∈ Bε (x) If (6) holds for all x ∗ ∈ ∂ϕ(x), then ϕ is said to be prox-regular at x. ¯ if for any Further, we say that ϕ is subdifferentially continuous at x¯ for x ∗ ∈ ∂ϕ(x) ε > 0 there exists δ > 0 such that |ϕ(x) − ϕ(x)| ¯ < ε whenever (x, x ∗ ) ∈ gph ∂ϕ ∩ (Bδ (x) × Bδ (x ∗ )). When this holds for all x ∗ ∈ ∂ϕ(x), the function ϕ is said to be subdifferentially continuous at x. It is easy to see that if ϕ is subdifferentially continuous at x¯ for x¯ ∗ , then the inequality “ϕ(x) < ϕ(x) + ε" in the definition of prox-regularity can be dropped. Extended-real-valued functions that are both proxregular and subdifferentially continuous are called continuously prox-regular. This is a major class of extended-real-valued functions in second-order variational analysis that is a common roof for particular collections of functions important in applications as, e.g., the so-called amenable functions, etc.; see [25, Chap. 13] and [3, 4]. The ϕ-attentive ε-localization of the subgradient mapping ∂ϕ around (x, x ∗ ) ∈ gph ∂ϕ is the set-valued mapping Tεϕ : X ⇒ X ∗ defined by    ϕ gph Tε := (x, x ∗ ) ∈ gph ∂ϕ  x − x < ε, |ϕ(x) − ϕ(x)| < ε and x ∗ − x ∗  < ε .

(7) If ϕ is an l.s.c. function, a localization can be taken with just ϕ(x) < ϕ(x) + ε. Now we start exploring the notion of epi-convergence of extended-real-valued functions on a Banach space X . More details and references can be found in [25] in finite dimensions and in [1, 5] in Banach spaces. Let    N∞ := N ⊂ IN  IN \ N finite ,    # N∞ := N ⊂ IN  N infinite .

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For a sequence {C k }k∈IN of subsets of X , the outer limit is the set  Lim sup C k : = x ∈ X k→∞  = x∈X

  N  ∃N ∈ N # , ∃x k ∈ C k (k ∈ N ) with x k − →x ∞    ∀V ∈ N (x), ∃N ∈ N # , ∀k ∈ N : C k ∩ V = ∅ , ∞

while the inner limit of {C k }k∈IN is defined by  Lim inf C k : = x ∈ X k→∞  = x∈X

  N  ∃N ∈ N∞ , ∃x k ∈ C k (k ∈ N ) with x k − →x    ∀V ∈ N (x), ∃N ∈ N∞ , ∀k ∈ N : C k ∩ V = ∅ ,

where N (x) denotes the collection of neighborhoods of x. The limit of the sequence {C k } exists if the outer and inner limit sets are equal, i.e., Lim C k := Lim sup C k = Lim inf C k . k→∞

k→∞

k→∞

We need the following simple observation. For a sequence of sets E k ⊂ X × IR that are epigraphs of some extended-real-valued functions, both the outer limit set Lim supk→∞ E k and inner limit set Lim inf k→∞ E k are also epigraphs of some functions. Indeed, if either set contains (x, α), then it also contains (x, α ) for all α ∈ [α, ∞). On the other hand, since both limit sets are closed, they intersect {x} × IR in a set which, unless empty, is a closed interval. Thusthecriteria for being an epigraph of some function is satisfied. For any sequence ϕk k∈N of extended-real-valued functions on X , the lower epi-limit, e-lim inf k→∞ ϕk , is the function having as its epigraph the outer limit of the sequence of sets epi ϕk , i.e., epi (e-lim inf ϕk ) := Lim sup(epi ϕk ). k→∞

k→∞

The upper epi-limit, e-lim supk→∞ ϕk , is the function having as its epigraph the inner limit of the sets epi ϕk , which is defined by epi (e-lim sup ϕk ) := Lim inf (epi ϕk ). k→∞

k→∞

Thus e- lim inf k→∞ ϕk ≤ e-lim supk→∞ ϕk in general. When these two functions coincide, it is said that the full limit epi-limit function e-limk→∞ ϕk exists, i.e., we have e- lim ϕk := e-lim inf ϕk = e- lim sup ϕk . k→∞

k→∞

k→∞ e

→ ϕ. It On this case, the functions ϕk epi-converge to ϕ, which is denoted by ϕk − k e k → ϕ ⇐⇒ epi ϕ → epi ϕ as k → ∞. The next characterization of is clear that ϕ −

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epi-convergence taken from [1, Proposition 1.14] is instrumental for deriving the main result of Sect. 3. Lemma 2.1 Let X be a Banach space, and let ϕk : X → IR as k ∈ IN and ϕ : X → IR be given functions. Then the following statements are equivalent: e

→ ϕ as k → ∞. (i) ϕk − (ii) At each point x ∈ X , we have the relationship ⎧ ⎨lim inf ϕk (x k ) ≥ ϕ(x) for every sequence x k → x, k→∞

⎩lim sup ϕk (x k ) ≤ ϕ(x) for some sequence x k → x. k→∞

To proceed further, along with standard definitions of lower semicontinuity of functions in the strong/norm topology of X and its sequential version, we recall their weak counterparts. A function ϕ : X → IR is weakly sequentially lower semicontinuous (weakly sequentially l.s.c.) at x ∈ dom ϕ if for any sequence {xk } which weakly converges to x, it holds that lim inf k→∞ ϕ(xk ) ≥ ϕ(x). We say that ϕ is weakly lower semicontinuous (weakly l.s.c.) at x if for any ε > 0 there exists a neighborhood U of x in the weak topology of X such that ϕ(x) ≥ ϕ(x) − ε for all x ∈ U. It is easily to see that the weak lower semicontinuity yields the weak sequential lower semicontinuity, while the reverse implication is not true in general; see, for instance, a counterexample in [14, Example 2.1]. On the other hand, if ϕ is weakly sequentially l.s.c. at x, then it is automatically l.s.c. at this point. The function ϕ is called lower semicontinuous (resp. weakly sequentially lower semicontinuous, weakly lower semicontinuous) around x if it possesses this property for all points in some neighborhood of x. Next we recall some notions related to monotonicity of set-valued mappings that are often called operators in this framework. The classical monotonicity notion is formulated as follows. An operator T : X ⇒ X ∗ is (globally) monotone on X if x1∗ − x2∗ , x1 − x2  ≥ 0 whenever (x1 , x1∗ ), (x2 , x2∗ ) ∈ gph T. It is called to be (globally) strongly monotone on X with modulus κ > 0 if x1∗ − x2∗ , x1 − x2  ≥ κx1 − x2 2 whenever (x1 , x1∗ ), (x2 , x2∗ ) ∈ gph T. A monotone (resp. strongly monotone) operator T is maximal monotone (resp. strongly maximal monotone) if gph T = gph S for any monotone operator S : X ⇒ X ∗ with gph T ⊂ gph S. We refer the reader to the monographs [2, 5, 20, 25] for various properties and applications of monotone and maximal monotone operators in finite and infinite dimensions. Note, in particular, that the graph of any maximal monotone mapping is nonempty and closed.

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The local monotonicity of set-valued mappings is naturally defined as follows; see, e.g., [25]. Definition 2.2 Let T : X ⇒ X ∗ , and let (x, x ∗ ) ∈ gph T . We say that: (i) T is locally monotone around (x, x ∗ ) if there are neighborhoods U of x and V of x ∗ with x1∗ − x2∗ , x1 − x2  ≥ 0 for all (x1 , x1∗ ), (x2 , x2∗ ) ∈ gph T ∩ (U × V ). (ii) T is locally monotone with respect to W ⊂ X × X ∗ if x1∗ − x2∗ , x1 − x2  ≥ 0 for all (x1 , x1∗ ), (x2 , x2∗ ) ∈ gph T ∩ W. (iii) T is locally strongly monotone with modulus κ > 0 around (x, y) if there are neighborhoods U of x and V of y satisfying the estimate x1∗ − x2∗ , x1 − x2  ≥ κx1 − x2 2 for all (x1 , x1∗ ), (x2 , x2∗ ) ∈ gph T ∩ (U × V ).

It is easy to observe the following robustness property: if T is locally monotone around (x, x ∗ ) with respect to an open neighborhood U × V , then we have the same property around any (x, x ∗ ) ∈ U × V . In this case, it is said that T is locally monotone relative to U × V . We now follow [19] to define the local maximal monotonicity of set-valued mappings. Definition 2.3 Let T : X ⇒ X ∗ , and let (x, x ∗ ) ∈ gph T . Then we say that: (i) T is locally maximal monotone around (x, x ∗ ) if there exist a neighborhood U × V of (x, x ∗ ) and a maximal monotone operator T : X ⇒ X ∗ such that gph T ∩ (U × V ) = gph T ∩ (U × V ). (ii) T is locally maximal monotone with respect to W ⊂ X × X ∗ if there exists a maximal monotone operator T : X ⇒ X ∗ such that gph T ∩ W = gph T ∩ W. (iii) T is locally strongly maximal monotone around (x, x ∗ ) if there is a neighborhood U × V of (x, x ∗ ) and a strongly maximal monotone operator T : X ⇒ X ∗ such that gph T ∩ (U × V ) = gph T ∩ (U × V ). Observe that if T is locally maximal monotone with respect to W ⊂ X × X ∗ , then it is also locally maximal monotone with respect to any subset Q ⊂ W .

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Finally in this section, we recall some geometric properties of Banach spaces; see, e.g., [9, 10, 18] for more details. A norm  ·  on a space X is called (Gâteaux) smooth if it is Gâteaux differentiable on X \ {0}. Equivalently, a norm on X is smooth if and only if the corresponding duality mapping J defined in (2) is single-valued on the whole X . Note that any separable Banach space admits an equivalent norm that is Gâteaux differentiable off the origin. The moduli of convexity and smoothness of X are defined, respectively, by  x + y ξ X (t) := inf 1 − 2

   x, y ∈ S X , x − y = t for all t ∈ [0, 2], and 

    1  x + y + x − y − 1  x ∈ S X , y = s for s > 0, ρ X (s) := sup 2 where S X := {x ∈ X | x = 1}. A Banach space X is said to be uniformly convex if ξ X (t) > 0. It is called 2-uniformly convex (resp. 2-uniformly smooth) if there exists a constant b > 0 such that ξ X (t) ≥ bt 2 (resp. ρ X (s) ≤ bs 2 ). It is shown in [8, Theorem 7 and Theorem 8, respectively] that any 2-uniformly convex space (resp. 2-uniformly smooth space) can be characterized via a lower (resp. upper) weak parallelogram law:   x + y2 + cx − y2 ≤ 2 x2 + y2 for all x, y ∈ X,

(8)

  x + y2 + cx − y2 ≥ 2 x2 + y2 for all x, y ∈ X.

(9)

In particular, a Banach space X is 2-uniformly convex (resp. 2-uniformly smooth) with constant b > 0 if and only if it satisfies condition (8) (resp. condition (9)) for some cb > 0. Therefore, any Hilbert space is simultaneously a 2-uniformly convex and 2-uniformly smooth Banach space. Moreover, it is obvious that any 2-uniformly convex space (resp. 2-uniformly smooth space) is uniformly convex (resp. uniformly smooth), and hence it is reflexive by the Milman-Pettis theorem. Some well-known non-Hilbert 2-uniformly convex (resp. 2-uniformly smooth) spaces are l p (μ) and L p (μ) with 1 < p < 2 (resp. with 2 < p < ∞). Recall [7] that in any reflexive Banach space, there exists an equivalent norm such that the corresponding duality mapping J from (2) is bijective and continuous with its inverse J −1 also being continuous. If in addition X is 2-uniformly convex, then it follows from [26] that the duality mapping is strongly monotone.

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3 Subdifferential Characterization of Variational Convexity In this section, we formulate the underlying notion of variational convexity for extended-real-valued functions on Banach spaces and establish its characterization in terms of the limiting subdifferential (4). The following definition is taken from Rockafellar [22], where it is formulated for functions on finite-dimensional spaces. Definition 3.1 An l.s.c. function ϕ : X → IR is called variationally convex at x¯ for ¯ if for some convex neighborhood U × V of (x, ¯ x¯ ∗ ) there exists a convex x¯ ∗ ∈ ∂ϕ(x) l.s.c. function ϕ  ≤ ϕ on U and ε > 0 such that we have the relationships ϕ and ϕ(x) =  ϕ(x) at the common elements (x, x ∗ ), (Uε × V ) ∩ gph ∂ϕ = (U × V ) ∩ gph ∂

where Uε := {x ∈ U | ϕ(x) < ϕ(x) + ε}. As demonstrated and discussed in [22] and also in [13], this notion is different from the usual local convexity of extended-real-valued functions. Now we present two lemmas on epi-convergence in minimization used in what follows. The first one is taken from [1, Proposition 2.9]. Lemma 3.2 Let X be a Banach space, and let ϕk : X → IR for k ∈ IN and ϕ : X → e IR be l.s.c. functions such that ϕk − → ϕ as k → ∞. Then we have inf ϕ ≥ lim sup(inf ϕk ).

(1)

k→∞

The second rather simple lemma reveals behavior of argminimum sets under epiconvergence of extended-real-valued functions defined on Banach spaces. Lemma 3.3 Let X be a Banach space, and let ϕk : X → IR for k ∈ IN and ϕ : X → e IR be l.s.c. functions. Suppose that ϕk − → ϕ as k → ∞ and that arg min ϕk = ∅ for all k ∈ IN as well as arg min ϕ = ∅. Then we have the inclusion Lim sup(arg min ϕk ) ⊂ arg min ϕ. k→∞

Therefore, for any choice of x k ∈ arg min ϕk , k ∈ IN, the sequence {x k } has all its cluster points belonging to arg min ϕ. If arg min ϕ consists of a unique point x, then x k → x as k → ∞. Proof 1 Pick x ∈ Lim sup(arg min ϕk ) and find subsequences {kn } of IN and x kn ∈ k→∞ e

→ ϕ, we have arg min ϕkn such that x kn → x as n → ∞. Since ϕk − lim sup(epi ϕk ) = epi ϕ. k→∞

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By 

 x, lim sup ϕkn (x kn ) = lim (x knl , ϕknl (x knl )) for some subsequence {knl } of {kn } l→∞

n→∞

and (x knl , ϕknl (x knl )) ∈ epi (ϕknl ) for all l ∈ IN, it follows that 

 x, lim sup ϕkn (x kn ) ∈ Lim sup(epi ϕk ). n→∞

k→∞

  Hence x, lim supn→∞ ϕkn (x kn ) ∈ epi ϕ, i.e., ϕ(x) ≤ lim sup ϕkn (x kn ). n→∞

This readily implies that ϕ(x) ≤ lim sup ϕkn (x kn ) = lim sup(inf ϕkn ) ≤ lim sup(inf ϕk ) ≤ inf ϕ, n→∞

n→∞

k→∞

where the last inequality is due to estimate (1) in Lemma 3.2. This gives us x ∈ arg min ϕ and therefore completes the proof of this lemma.  The next result provides a major characterization of variational convexity of extended-real-valued functions defined on Banach spaces. It is a far-going generalization of the finite-dimensional characterization given in [22, Theorem 1, (c)⇐⇒(b)]. Theorem 3.4 Let ϕ : X → IR be an l.s.c. function defined on a Banach space X , and let x ∗ ∈ ∂ϕ(x) be a basic subgradient from (4). Consider the following assertions: (i) ϕ is variationally convex at x¯ for x ∗ . (ii) There exists a convex neighborhood U × V of (x, x ∗ ) along with ε > 0 such that we have

    (u, u ∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ =⇒ ϕ(x) ≥ ϕ(u) + u ∗ , x − u for all x ∈ U, (2)

where Uε is taken from Definition 3.1. Then implication (i)=⇒(ii) holds in general Banach spaces. The reverse implication is satisfied if X is a reflexive space and if ϕ is weakly sequentially l.s.c. around x. Proof 2 First we verify implication (i)=⇒(ii) in any Banach space X . Assuming the variational convexity of ϕ at x for x ∗ , find a convex neighborhood U × V of (x, x ∗ ) and a convex l.s.c. function ϕ  ≤ ϕ on U such that ϕ and ϕ(x) =  ϕ(x) at the common elements (x, x ∗ ) (Uε × V ) ∩ gph ∂ϕ = (U × V ) ∩ gph ∂

for some ε > 0. Without loss of generality, suppose that ϕ  is convex on X . For any ϕ, and therefore (u, u ∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ we get (u, u ∗ ) ∈ gph ∂

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ϕ (x) ≥ ϕ (u) + u ∗ , x − u whenever x ∈ U. Since ϕ (x) ≤ ϕ(x) and ϕ (u) = ϕ(u), it follows that ϕ(x) ≥ ϕ(u) + u ∗ , x − u for all x ∈ U , which justifies implication (i)=⇒(ii). To verify next the reverse implication (ii)=⇒(i), observe first that having (ii) allows us to shrink U and V if necessary so that U ⊂ B(x, r1 ) and V ⊂ B(x ∗ , r2 )  : X → IR by with r1 < ε/(x ∗  + r2 ). Define the function ϕ    ϕ (x) := sup ϕ(u) + u ∗ , x − u  (u, u ∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ , x ∈ X. (3) Since ϕ  is the supremum of a collection of affine functions, it is l.s.c. and convex on the whole space X . We clearly have the estimate ϕ (x) ≥ ϕ(x) + x ∗ , x − x > −∞ whenever x ∈ X. Moreover, it follows from (2) and the construction of ϕ  that ϕ  ≤ ϕ on U . We split the remaining proof of (ii)=⇒(i) into the six steps/claims as follows. Claim 1: In any Banach space X , the inclusion (x, x ∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ is ϕ together with ϕ (x) = ϕ(x) for equivalent to having (x, x ∗ ) ∈ (U × V ) ∩ gph ∂ any l.s.c. function ϕ : X → IR. To justify this claim, consider (x, x ∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ and check first that ϕ (x) = ϕ(x). Indeed, ϕ(x) can be equivalently written as ϕ(x) = ϕ(x) + x ∗ , x − x with (x, x ∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ, which implies that ϕ (x) ≥ ϕ(x). Combining the latter with the inequality ϕ  ≤ ϕ on ϕ, deduce from U , we arrive at ϕ (x) = ϕ(x). To check further that (x, x ∗ ) ∈ gph ∂ (x, x ∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ that for any x  ∈ X it follows that (x) = x ∗ , x  − x + ϕ(x) ≤ ϕ (x  ), x ∗ , x  − x + ϕ where the last inequality is a consequence of (3). This tells us that x ∗ , x  − x ≤ ϕ (x  ) − ϕ (x) for all x  ∈ X , i.e., x ∗ ∈ ∂ ϕ(x). ϕ To verify the reverse implication in Claim 1, pick (x, x ∗ ) ∈ (U × V ) ∩ gph ∂ such that ϕ (x) = ϕ(x). We aim to show that (x, x ∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ. By x ∗ ∈ ∂ ϕ(x), we have (x) ≤ ϕ (x  ) for all x  ∈ X. x ∗ , x  − x + ϕ It follows from ϕ  ≤ ϕ on U that x ∗ , x  − x + ϕ(x) ≤ ϕ(x  ) whenever x  ∈ U, and therefore we have the limiting condition

(4)

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lim inf x →x

ϕ(x  ) − ϕ(x) − x ∗ , x  − x ≥ 0, x  − x

 and hence x ∗ ∈ ∂ϕ(x). It remains to check that x ∈ Uε , which means that x ∗ ∈ ∂ϕ(x) i.e., ϕ(x) < ϕ(x) + ε. Indeed, it follows from (4) and the choice of U and V above that ϕ(x) ≤ ϕ(x) − x ∗ , x − x ≤ ϕ(x) + x ∗  · x − x ≤ ϕ(x) + (x ∗ − x ∗  + x ∗ ) · x − x ≤ ϕ(x) + (r2 + x ∗ )r1 < ϕ(x) + ε, which completes the proof of Claim 1. To proceed further, select a bounded, closed, and convex set  ⊂ U containing x as an interior point and such that ϕ is weakly sequentially l.s.c. at any point in . ϕ, define the auxiliary l.s.c. function ψx,x ∗ by For (x, x ∗ ) ∈ gph ∂ 1 (x) − x ∗ , y − x + y − x2 + δ (y), ψx,x ∗ (y) := ϕ(y) − ϕ 2

y ∈ X,

(5)

via the indicator function δ (x) of  equal to 0 if x ∈  and ∞ otherwise. Claim 2: Let X be a Banach space, and let ψx,x ∗ be taken from (5). Then for ϕ and any y ∈ X , we have ψx,x ∗ (y) ≥ 0. Furthermore, it follows any (x, x ∗ ) ∈ gph ∂ that argmin ψx,x ∗ = {x} and min ψx,x ∗ = 0. ϕ ensures that x ∗ , y − x + ϕ (x) ≤ ϕ (y) ≤ ϕ(y) Indeed, picking (x, x ∗ ) ∈ gph ∂ whenever y ∈ U . Since  ⊂ U , it follows that ψx,x ∗ (y) ≥ 0 for all y ∈ X , and that ψ vanishes if and only if ⎧ ⎪ ⎨ y = x, y ∈ , ⎪ ⎩ ϕ(y) = ϕ (x). Applying this to (x, x ∗ ) gives us arg min ψx,x ∗ = {x} and min ψx,x ∗ = 0 as stated in Claim 2. Claim 3: LetX be a reflexive Banach space, and let ϕ : X → IR be weakly sequentially l.s.c. around x. Then each function ψx,x ∗ achieves its global minimum onX . Since  is closed and convex, this set is weakly closed by the classical Mazur theorem, and hence it is weakly sequentially closed. On the other hand, we know that the function x ∗ , · − x is weakly continuous, while  · −x2 is weakly sequentially l.s.c. around x. Also, the indicator function δ (·) is weakly sequentially l.s.c. at any point in . Combining this with the imposed weakly sequentially l.s.c. property of ϕ tells us that the function ψx,x ∗ is weakly sequentially l.s.c. on the bounded and weakly sequentially closed subset  of the reflexive Banach space X . Thus the generalized Weierstrass theorem in this setting (see, for instance, [15, Theorem 7.3.4]) ensures

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that each ψx,x ∗ attains its minimum on , which is actually a global minimum on X due to the construction of ψx,x ∗ in (5). We are done with the proof of Claim 3. Claim 4: LetX be a Banach space, and let ϕ : X → IR be l.s.c. onX . Then the function ϕ, in the sense that ψx,x ∗ depends epicontinuously on (x, x ∗ ) with respect to gph ∂ gph ∂ϕ e ∗ ∗ → ψx,x ∗ as k → ∞. the convergence (xk , xk ) −−−−→ (x, x ) implies that ψxk ,xk∗ − gph ∂ϕ To justify this claim, take a sequence (xk , xk∗ ) −−−−→ (x, x ∗ ) as k → ∞ and verify e → ψx,x ∗ by using its characterization given in Lemma 2.1. the epi-convergence ψxk ,xk∗ −  is l.s.c. and convex To this end, pick any y ∈ X and any sequence yk → y. Since ϕ on X , it is subdifferentially continuous at x ∈ dom ϕ ; see, e.g., [25, Example 13.30] gph ∂ϕ which proof works in arbitrary Banach spaces. Combining this with (xk , xk∗ ) −−−−→ (xk ) → ϕ (x) as k → ∞. Therefore, we readily (x, x ∗ ) gives us the convergence ϕ get the relationships lim inf ψxk ,xk∗ (yk ) ≥ lim inf ϕ(yk ) + lim inf δ (yk ) − lim  ϕ(xk ) k→∞ k→∞ k→∞ k→∞  1 1 + lim yk − xk 2 − xk∗ , yk − xk  ≥ ϕ(y) + δ (y) −  ϕ(x) + y − x2 − x ∗ , y − x k→∞ 2 2 = ψx,x ∗ (y).

On the other hand, choosing the constant sequence z k := y tells us that lim sup ψxk ,xk∗ (y) = lim sup ϕ(y) + lim sup δ (y) − lim  ϕ(xk ) k→∞ k→∞ k→∞ k→∞ 1 1 y − xk 2 − xk∗ , y − xk  = ϕ(y) + δ (y) −  + lim ϕ(x) + y − x2 − x ∗ , y − x k→∞ 2 2 = ψx,x ∗ (y), e

which ensures that ψxk ,xk∗ − → ψx,x ∗ as k → ∞ and thus verifies the claim. Claim 5: LetX be a reflexive Banach space. Suppose that the sequence (xk , xk∗ ) ∈ (U × V ) ∩ gph ∂ ϕ converges to some (x, x ∗ ) as k → ∞ and that the function ϕ is weakly sequentially l.s.c. around x. Then for each k ∈ IN sufficiently large, there exists a pair (x˜k , x˜k∗ ) ∈ (Uε × V ) ∩ gph ∂ϕ such that ϕ (x˜k ) = ϕ(x˜k ) and xk∗ − x˜k∗ ∈ J (x˜k − xk ). e

Indeed, it follows from Claim 4 that ψxk ,xk∗ − → ψx,x ∗ as k → ∞. Taking into account that arg min ψxk ,xk∗ = ∅ by Claim 3 and employing Lemma 3.3 allow us to construct a sequence {x˜k } such that x˜k → x as k → ∞ and that x˜k ∈ arg min ψxk ,xk∗ ∩ int  for all large k ∈ IN. Lemma 3.2 gives us the relationships   0 = min ψx,x ∗ ≥ lim sup(inf ψxk ,xk∗ ) = lim sup ψxk ,xk∗ (x˜k ) , k→∞

k→∞

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which ensure that ψxk ,xk∗ (x˜k )  0 due to ψxk ,xk∗ (x˜k ) ≥ 0 for all k ∈ IN. Thus for sufficiently large k ∈ IN, we have the conditions 1 ε (xk ) = ψxk ,xk∗ (x˜k ) + xk∗ , x˜k − xk  − x˜k − xk 2 < . ϕ(x˜k ) − ϕ 2 2 The latter clearly implies the estimate ε 2  is subdifferentially continuous at x) ≤ϕ (x) + ε (since ϕ

ϕ(x˜k ) < ϕ (xk ) +

= ϕ(x) + ε, which tells us that x˜k ∈ Uε for sufficiently large k. Remembering that x˜k is a minimizer of ψxk ,xk∗ on X , we get by the subdifferential Fermat rule (see, e.g., [16, Proposition 1.114]) that 0 ∈ ∂ψxk ,xk∗ (x˜k ). Since ϕ is l.s.c. while all other functions in (5) are locally Lipschitzian around x due to the construction of ψxk ,xk∗ , and since any reflexive Banach space is Asplund, we apply to ∂ψxk ,xk∗ (x˜k ) the semi-Lipschitzian subdifferential sum rule from [16, Theorem 2.33] to obtain 0 ∈ ∂ψxk ,xk∗ (x˜k ) ⊂ ∂ϕ(x˜k ) − xk∗ + J (x˜k − xk ) for all large k ∈ IN. Observe that ∂( · −xk 2 )(x˜k ) = J (x˜k − xk ) via the duality mapping (2) and that ∂δ (x˜k ) = {0} due to the choice of x and the convergence x˜k → x. Thus there exist x˜k∗ ∈ ∂ϕ(x˜k ) and j (x˜k − xk ) ∈ J (x˜k − xk ) such that x˜k∗ = xk∗ − j (x˜k − xk ). Since (xk , xk∗ ) → (x, x ∗ ) and x˜k → x, we also have x˜k − xk → 0 as k → ∞. By the fact that  j (x˜k − xk ) = x˜k − xk , this gives us j (x˜k − xk ) → 0, and so x˜k∗ → x ∗ as k → ∞. In summary, for all k sufficiently large, there are (x˜k , x˜k∗ ) ∈ gph ∂ϕ such that x˜k ∈ Uε and (x˜k , x˜k∗ ) → (x, x ∗ ) as k → ∞. Therefore, (x˜k , x˜k∗ ) ∈ (Uε × ϕ and ϕ (x˜k ) = ϕ(x˜k ) for large k due V ) ∩ gph ∂ϕ. Consequently, (x˜k , x˜k∗ ) ∈ gph ∂ to Claim 1. This verifies the formulated assertions. Claim 6: In the setting of Claim 5, we have x˜k = xk for all large k ∈ IN. Indeed, the subgradient mapping ∂ ϕ is monotone due to the properness and convexity ϕ due to Claim 5, we get of ϕ . Since both pairs (xk , xk∗ ) and (x˜k , x˜k∗ ) belong to gph ∂ that 0 ≤ xk∗ − x˜k∗ , xk − x˜k  = −xk − x˜k 2 , and thus x˜k = xk , which verifies this claim. First we verify Unifying the above claims with shrinking the neighborhoods U ϕ we have ϕ (x) = and V if necessary tell us that for all (x, x ∗ ) ∈ (U × V ) ∩ gph ∂ ϕ(x). This completes the proof of implication (ii)=⇒(i) and hence of the whole theorem. 

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4 Characterizations of Variational Convexity via Local Monotonicity and Moreau Envelopes This section provides new characterizations of variational convexity of extendedreal-valued l.s.c. functions on Banach spaces. The first type of characterizations involves local monotonicity and local maximal monotonicity of limiting subgradient mappings, while the second type characterizes variational convexity of functions via the standard local convexity of their Moreau envelopes. Given an l.s.c. function ϕ : X → IR on a Banach space X together with a parameter λ > 0, the Moreau envelope eλ ϕ : X → IR and the proximal mapping Pλ ϕ : X ⇒ X associated with ϕ and λ are defined, respectively, by  1 2 w − x , x ∈ X, eλ ϕ(x) := inf ϕ(w) + w∈X 2λ

(1)

 1 Pλ ϕ(x) := argminw∈X ϕ(w) + w − x2 , x ∈ X. 2λ

(2)

Prior to establishing the main characterizations of this section, we present several lemmas, which are of their own interest while being instrumental for the proof of the major results below. The first lemma provides a description of the prox-boundedness (5) of extended-real-valued functions via their Moreau envelopes (1). Lemma 4.1 Let ϕ : X → IR be an l.s.c. function defined on a Banach space X . Then the following assertions are equivalent: (i) ϕ is prox-bounded. (ii) There exist λ > 0 and x ∈ X such that eλ ϕ(x) > −∞. (iii) There exists λ0 > 0 such that eλ ϕ(x) > −∞ for any 0 < λ < λ0 and for any x ∈ X. Proof 3 We begin with verifying implication (i)=⇒(ii). The prox-boundedness of ϕ gives us α, β ∈ IR and x ∈ X such that ϕ(y) ≥ αy − x2 + β for all y ∈ X. Then for any λ > 0 satisfying α + ϕ(y) +

1 ≥ 0, we get 2λ

1 1  y − x2 ≥ α + y − x2 + β whenever y ∈ X. 2λ 2λ

Taking the infimum of both sides above with respect to y ∈ X brings us to eλ ϕ(x) ≥ β > −∞, which therefore verifies assertion (ii). Suppose next that (ii) holds. Then denote

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   λ0 := sup λ > 0  ∃x ∈ X and eλ ϕ(x) > −∞ and deduce from (ii) that λ0 > 0. Consider any 0 < λ < λ0 and any x ∈ X . Since λ < λ0 , there exists λ¯ > λ with x ∈ X satisfying eλ¯ ϕ(x) > −∞. For y ∈ X , we clearly have that ϕ(y) +

1 1 1 1 y − x2 = ϕ(y) + y − x2 − y − x2 + y − x2 2λ 2λ 2λ¯ 2λ¯ 1 1 y − x2 − ≥ eλ¯ ϕ(x) + (y − x + x − x)2 2λ 2λ¯   1 1 1 1 · y − x2 − y − x · x − x − − = eλ¯ ϕ(x) + x − x2 . 2λ 2λ¯ 2λ¯ λ¯

1 y − x2 is bounded Since eλ¯ ϕ(x) > −∞ and λ¯ > λ, the expression ϕ(y) + 2λ below when y varies in X . This gives us the conditions  1 y − x2 > −∞, eλ ϕ(x) = inf ϕ(y) + y∈X 2λ which verify (iii). The remaining implication (iii)=⇒(i) is trivial, and thus we are done.  Given x ∗ ∈ X ∗ and λ > 0, define the x ∗ -Moreau envelope for ϕ : X → IR by ∗ eλx ϕ(x)

 := inf

w∈X

1 2 ϕ(w) − x , w + w − x , x ∈ X, 2λ ∗

(3)

and the x ∗ -proximal mapping associated with ϕ by ∗ Pλx ϕ(x)

 1 ∗ 2 w − x , x ∈ X. := argminw∈X ϕ(w) − x , w + 2λ

(4)

It is easy to see that the x ∗ -Moreau envelope for ϕ at x ∈ X is the standard Moreau envelope for the tilted function ϕ(·) − x ∗ , · at the same point. Therefore, we have the following relationships between x ∗ -Moreau envelopes (resp. x ∗ -proximal mappings) and standard Moreau envelopes (resp. proximal mappings) in the setting of Hilbert spaces: λ ∗ 2 ∗ x  and Pλx ϕ(x) = Pλ ϕ(x + λx ∗ ). 2 (5) This result can be found in [11, Lemma 2.2] which proof works for functions on Hilbert spaces. The next lemma provides characterizations of fixed points of the x ∗ -proximal mappings defined in (4). Recall from [25, Definition 8.45] that x ∗ ∈ X ∗ is a proximal subgradient of ϕ at x ∈ dom ϕ if there exist numbers r > 0 and ε > 0 such that for ∗

eλx ϕ(x) = eλ ϕ(x + λx ∗ ) − x ∗ , x −

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all x ∈ B(x, ε) we have r ϕ(x) ≥ ϕ(x) + x ∗ , x − x − x − x2 . 2

(6)

Lemma 4.2 Consider an l.s.c. function ϕ : X → IR with x ∈ dom ϕ, and x ∗ ∈ X ∗ . Then the following statements are equivalent: (i) ϕ is prox-bounded, and x ∗ is a proximal subgradient of ϕ at x. ∗ (ii) Pλx ϕ(x) = {x} for some λ > 0. ∗ (iii) Pλx ϕ(x) = {x} for all λ > 0 sufficiently small. Proof 4 Assume that (i) holds. To verify (ii), take x ∗ as a proximal subgradient of ϕ at x and then find r, ε > 0 such that (6) is satisfied. We intend to show that (6) holds for all x ∈ X with a certain modification of r > 0. Since ϕ is prox-bounded, Lemma 4.1 implies that there exists λ > 0 with eλ ϕ(x) > −∞. On the other hand, choosing λ¯ > 0 to be sufficiently small guarantees that the estimate ϕ(x) + x ∗ , x − x −

1 1 x − x2 ≤ ϕ(x) x − x2 ≤ eλ ϕ(x) − 2λ 2λ¯

(7)

¯ and deduce from holds for all x satisfying x − x > ε. Denote r := 2 max{r, 1/λ} (6) and (7) the fulfillment of the strict inequality r ϕ(x) − x ∗ , x + x − x2 > ϕ(x) − x ∗ , x for all x = x. 2 The latter is equivalent to saying that Pλ¯x (x) = {x}, where λ¯ := 1/r , and hence (ii) is verified. Implication (ii)=⇒(iii) is trivial. Finally, suppose that (iii) holds and deduce from ∗ ∗ Pλx ϕ(x) = {x} that eλx ϕ(x) > −∞, which yields the prox-boundedness of ϕ by Lemma 4.1. The assertion that x ∗ is a proximal subgradient of ϕ at x is implied by 1 the fact that x is a global minimizer of ϕ(·) − x ∗ , · +  · −x2 . Thus we arrive 2λ at (i) and complete the proof of the lemma.  ∗

The following lemma is a combination of the results taken from [4, Theorems 5.3 and 5.5]. Lemma 4.3 Let X be a 2-uniformly convex space which norm is Gâteaux differentiable on X \ {0}, and let ϕ : X → IR be an l.s.c., prox-bounded, and prox-regular function at x for x ∗ ∈ ∂ϕ(x). Then there exist λ0 > 0 and ε > 0 such that for any ∗ positive number λ ≤ λ0 , there is some neighborhood Uλ of x on which eλx ϕ is C 1 smooth with ∗ ∗ ∇eλx ϕ = λ−1 J ◦ (I − Pλx ϕ), ∗

while the proximal mapping Pλx ϕ is single-valued, continuous, and admits the representation

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−1   ∗ Pλx ϕ(u) = I + λJ −1 ◦ Tεϕ − x ∗ (u) for all u ∈ Uλ , ε-localization of ∂ϕ around (x, x ∗ ) taken from (7), and where Tεϕ is the ϕ-attentive   ϕ ∗ ϕ where the mapping Tε − x (x) := Tε (x) − x ∗ is also single-valued on Uλ . The next result reveals important properties of the duality mapping (2), which are instrumental in the proof of the main result below. Lemma 4.4 Let X be a uniformly convex Banach space which norm is Gâteaux differentiable at nonzero points of X . Then the corresponding duality mapping J is single-valued and norm-to-norm continuous on the whole space X . Proof 5 Fix an arbitrary point x ∈ X and take a sequence xk → x as k → ∞. By the Milman-Pettis theorem, the space X is reflexive. Due to the reflexivity of X and the assumed smoothness of the norm  · , it follows from [9, Corollary 1.4] that the dual space X ∗ is strictly convex, i.e., its closed unit ball is a strictly convex set. Moreover, [9, Corollary 1.5] tells us that J is single-valued and norm-to-weak∗ w∗

continuous. This implies that J (xk ) −→ J (x) as k → ∞. Since X is reflexive, we w → J (x). Furthermore, the convergence xk  → x and defiactually have J (xk ) − nition (2) of the duality mapping J ensure that J (xk ) → J (x). Finally, we get by [6, Proposition 3.32] that J (xk ) → J (x) as k → ∞, which thus completes the proof.  Before deriving the major relationships for variational convexity of functions defined on Banach spaces, we formulate the modified notions of local monotonicity and local maximal monotonicity of subgradient mappings by following the finitedimensional pattern of [22]. Definition 4.5 Given an l.s.c. function ϕ : X → IR defined on a Banach space X , we say that the subgradient mapping ∂ϕ : X ⇒ X ∗ is ϕ-locally monotone (ϕ-locally maximal monotone) around (x, ¯ x¯ ∗ ) ∈ gph ∂ϕ if there exist a neighborhood U × V ∗ of (x, ¯ x¯ ) and a number ε > 0 such that the mapping ∂ϕ is locally monotone (locally maximal monotone) with respect to Uε × V , where Uε is taken from Definition 3.1. Here are the aforementioned major results on variational convexity of extendedreal-valued l.s.c. functions defined on Banach spaces. The equivalence between assertions (i), (ii), and (iii) of the next theorem was established in [22] in finite-dimensional spaces. In [13], the variational convexity of an l.s.c. prox-bounded function ϕ at x for v¯ ∈ ∂ϕ(x) was characterized in finite dimensions by the local convexity of its Moreau envelope eλ ϕ from (1) around x + λv¯ for small λ > 0. Our theorem below offers more variety in infinite dimensions by involving geometric properties of Banach spaces. Note, in particular, that—outside of Hilbert spaces—we now replace in assertion (iv) below the local convexity of the Moreau envelope eλ ϕ around x + λv¯ by the local convexity of the tilted x ∗ -Moreau envelope from (3) around x. Theorem 4.6 Let X be a Banach space, and let ϕ : X → IR be an l.s.c. function with x ∗ ∈ ∂ϕ(x). Consider the following assertions:

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(i) (ii) (iii) (iv)

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ϕ is variationally convex at x for x ∗ . ∂ϕ is ϕ-locally maximal monotone around (x, x ∗ ). ∂ϕ is ϕ-locally monotone around (x, x ∗ ). ∗ The x ∗ -Moreau envelope eλx ϕ is locally convex around x for all small λ > 0.

Then we have implications (i)=⇒(ii)=⇒(iii) in general Banach spaces. Implication (iii)=⇒(iv) holds when X is a 2-uniformly convex space which norm is smooth, while ϕ is prox-bounded and prox-regular at x for x ∗ . Finally, implication (iv)=⇒(i) is satisfied if in addition the function ϕ is weakly sequentially l.s.c. around x. Proof 6 First we verify implication (i)=⇒(ii) under the general assumptions on ϕ and X . It follows from the variational convexity of ϕ at x for x ∗ that there exist  : X → IR ε > 0, a neighborhood U × V of (x, x ∗ ), and an l.s.c. convex function ϕ such that (8) gph ∂ ϕ ∩ (U × V ) = gph ∂ϕ ∩ (Uε × V ). The classical result of [21, Theorem A] applied to ϕ  on a Banach space X tells us that the subgradient mapping ∂ ϕ is (globally) monotone on X . Combining the latter with (8) implies that the mapping ∂ϕ is locally maximal monotone with respect to Uε × V , which thus justifies (ii) by Definition 4.5. The next implication (ii)=⇒(iii) is trivial. Before verifying implications (iii)=⇒(iv) and (iv)=⇒(i), we deduce from Lemma 4.3 that if X is smooth and 2-uniformly convex and ϕ is prox-bounded and prox-regular, then there exist λ0 > 0, γ > 0, and a ϕ-attentive γ-localization Tγϕ : X ⇒ X ∗ given by  Tγϕ (x)

:=

  x ∗ ∈ ∂ϕ(x) x ∗ − x ∗  < γ if x − x < γ and ϕ(x) < ϕ(x) + γ, (9) ∅ otherwise ∗

such that for any λ ∈ (0, λ0 ) there is a neighborhood Uλ of x on which eλx ϕ is of ∗ class C 1 , that the x ∗ -proximal mapping Pλx ϕ is single-valued and continuous on Uλ with  −1 ∗ (10) Pλx ϕ(u) = I + λJ −1 ◦ (Tγϕ − x ∗ ) (u) for all u ∈ Uλ , and that we have the gradient representation ∗

∗

∇eλx ϕ = λ−1 J ◦ (I − Pλx ϕ).

(11)

To justify now implication (iii)=⇒(iv), suppose that ∂ϕ is ϕ-locally monotone around (x, x ∗ ) and then find ε > 0 and r > 0 such that the subgradient mapping ∂ϕ is monotone with respect to the set W ε := Br (x)ε × Br (x ∗ ), where    Br (x)ε := x ∈ Br (x)  ϕ(x) < ϕ(x) + ε .

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Since ϕ is prox-bounded and x ∗ is a proximal subgradient of ϕ at x by the assumed prox-regularity, it follows from Lemma 4.2 that ∗

∗

Pλx ϕ(x) = x and eλx ϕ(x) = ϕ(x) − x ∗ , x.

(12)

Using the uniform convexity of X together with the smoothness of its norm  · , we deduce from Lemma 4.4 that J is norm-to-norm continuous. Considering further the function   1 ∗ ∗ ∗ Pλx (·) − ·2 , x ∈ Uλ ψλ := eλx ϕ(·) + x ∗ , Pλx ϕ(·) − 2λ and invoking (11) and (12) give us the equalities 1 ∗ P x (x) − x2 = ϕ(x), 2λ λ   ∗ ∗ ∇eλx ϕ(x) = λ−1 J ◦ x − Pλx ϕ(x) = 0.

∗

∗

ψλ (x) = eλx ϕ(x) + x ∗ , Pλx ϕ(x) −

∗

∗

It follows from (12) and the continuity of the mappings ψλ , ∇eλx ϕ, and Pλx ϕ around x that   ∗ ∗   Pλx ϕ(x) ∈ Br (x), ψλ (x) < ϕ(x) + ε, and ∇eλx ϕ(x) < r for all x ∈ U (13) when a neighborhood U ⊂ Uλ of x is sufficiently small. The definitions in (3), (4) ∗ and the above construction of ψλ ensure that ψλ (x) = ϕ(Pλx ϕ(x)) for all x ∈ U . ∗ Pick xi ∈ U and denote yi := Pλx ϕ(xi ) for i = 1, 2. Employing the representations in (10) and (11) and remembering that the duality mapping J is bijective in our setting bring us to ∗

λ−1 J (xi − yi ) + x ∗ ∈ Tγϕ (yi ) and ∇eλx ϕ(xi ) = λ−1 J (xi − yi ), i = 1, 2. (14) It follows from (13) that yi ∈ Br (x), λ−1 J (xi − yi ) + x ∗ ∈ Br (x ∗ ), and  ∗  ϕ(yi ) = ϕ Pλx ϕ(xi ) = ψλ (xi ) < ϕ(x) + ε, i = 1, 2, ∗

which leads us to (yi , ∇eλx ϕ(xi ) + x ∗ ) ∈ W ε ∩ gph ∂ϕ. Observing by (14) that xi = ∗ yi + λJ −1 (∇eλx ϕ(xi )), we get by the local monotonicity of ∂ϕ with respect to W ε and the global monotonicity of J −1 on X ∗ that 

   ∗ ∗ ∗ ∗ ∇eλx ϕ(x1 ) − ∇eλx ϕ(x2 ), x1 − x2 = ∇eλx ϕ(x1 ) − ∇eλx ϕ(x2 ), y1 − y2

     ∗ ∗ ∗ ∗ +λ ∇eλx ϕ(x1 ) − ∇eλx ϕ(x2 ), J −1 ∇eλx ϕ(x1 ) − J −1 ∇eλx ϕ(x2 ) ≥ 0.

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∗

This tells us that ∇eλx ϕ is locally monotone around x. Utilizing finally the result of [12, Theorem 4.1.4] which proof works also for a general Banach space, we arrive ∗ at the local convexity of eλx ϕ around x, and thus verify (iv). Let us next proceed with the proof of implication (iv)=⇒(i). Assume that the ∗ x ∗ -Moreau envelope eλx ϕ is locally convex around x for all λ > 0 sufficiently small, that X is smooth and 2-uniformly convex, and that ϕ is prox-bounded, prox-regular, and weakly sequentially l.s.c. around x. Fixing λ ∈ (0, λ0 ), suppose without loss ∗ of generality that eλx ϕ is convex on Uλ . Utilizing the first-order characterization of C 1 -smooth convex functions in [12, Theorem 4.1.1], which proof holds in general Banach spaces, gives us the inequality ∗

∗

∗

eλx ϕ(x) ≥ eλx ϕ(u) + ∇eλx ϕ(u), x − u for all x, u ∈ Uλ . Select convex neighborhoods U ⊂ Uλ of x and V of x ∗ such that U ⊂ Bγ (x), V ⊂ Bγ (x ∗ ), and x + λJ −1 (x ∗ − x ∗ ) ∈ Uλ whenever (x, x ∗ ) ∈ U × V. Theorem 3.4 reduces justifying the variational convexity of ϕ at x for x ∗ to verifying that ϕ(x  ) ≥ ϕ(x) + x ∗ , x  − x for all x  ∈ U, (x, x ∗ ) ∈ (Uγ × V ) ∩ gph ∂ϕ, (15) where Uγ = {x ∈ U | ϕ(x) < ϕ(x) + γ}. Pick x  ∈ U , (x, x ∗ ) ∈ (Uγ × V ) ∩ gph ∂ϕ and deduce from (9) that (x, x ∗ ) ∈ gph Tγϕ . Employing (10) and (11) gives us the equalities     ∗ ∗ x = Pλx ϕ x + λJ −1 (x ∗ − x ∗ ) and ∇eλx ϕ x + λJ −1 (x ∗ − x ∗ ) = x ∗ − x ∗ , which yield in turn to the relationships   1  x  − x  + λJ −1 (x ∗ − x ∗ ) 2 ϕ(x  ) − x ∗ , x   + 2λ   ∗ ≥ eλx ϕ x  + λJ −1 (x ∗ − x ∗ )       ∗ ∗ ≥ eλx ϕ x + λJ −1 (x ∗ − x ∗ ) + ∇eλx ϕ x + λJ −1 (x ∗ − x ∗ ) , x  − x     1  ∗ ∗ 2 −1 ∗  + x ∗ − x ∗ , x  − x. x − x + λJ (x − x ) = ϕ(x) − x , x + 2λ After the simplification of the above, we arrive at the estimate ϕ(x  ) ≥ ϕ(x) + x ∗ , x  − x, which verifies (15) and thus completes the proof of the theorem.



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If x is a stationary point of ϕ, i.e., x ∗ = 0, we get the immediate consequence of Theorem 4.6. Corollary 4.7 Let X be a 2-uniformly convex Banach space which norm is Gâteaux differentiable at nonzero points, and let ϕ : X → IR be an l.s.c. and prox-bounded function with x ∈ dom ϕ and 0 ∈ ∂ϕ(x). Consider the following assertions: (i) ϕ is variationally convex at x for 0. (ii) ϕ is prox-regular at x for 0, and the Moreau envelope eλ ϕ is locally convex around x for all small λ > 0. Then we have implication (i)=⇒(ii) under the general assumptions made, while the reverse implication (ii)=⇒(i) holds provided that ϕ is weakly sequentially l.s.c. around x. The last result of this section establishes a characterization of variational convexity for functions defined on Hilbert spaces, where we can replace the x ∗ ¯ where Moreau envelope (3) around x by its standard counterpart (1) around x + λv, x ∗ ∈ ∂ϕ(x) ⊂ X ∗ is replaced by v¯ ∈ ∂ϕ(x) ⊂ X due to X = X ∗ . The corollary below is a Hilbert space extension of the finite-dimensional characterization obtained in [13, Theorem 3.2]. Corollary 4.8 Let X be a Hilbert space, and let ϕ : X → IR be an l.s.c. and proxbounded function with x¯ ∈ dom ϕ and v¯ ∈ ∂ϕ(x). ¯ Consider the following assertions: ¯ (i) ϕ is variationally convex at x for v. ¯ and the Moreau envelope eλ ϕ is locally convex (ii) ϕ is prox-regular at x for v, around x¯ + λv¯ for all small λ > 0. Then implication (i)=⇒(ii) is satisfied under the general assumptions imposed, while the reverse implication (ii)=⇒(i) holds if in addition ϕ is weakly sequentially l.s.c. around x. Proof 7 To deduce this corollary from Theorem 4.6, it suffices to show that the local convexity of the v-Moreau ¯ envelope eλv¯ ϕ around x is equivalent to the local convexity ¯ Indeed, we get from (5) that of the standard one eλ ϕ around x + λv. ¯ − v, ¯ x − eλv¯ ϕ(x) = eλ ϕ(x + λv)

λ v ¯ 2 for all x ∈ X. 2

(16)

Defining ψv¯ (x) := x + λv¯ for x ∈ X , the equality in (16) can be rewritten as ¯ x − eλv¯ ϕ(x) = (eλ ϕ ◦ ψv¯ )(x) − v,

λ v ¯ 2 , x ∈ X. 2

Since ψv¯ is an affine mapping while v, ¯ · is a convex function, the local convexity of eλ ϕ around x + λv¯ yields the same property of eλv¯ ϕ around x. On the other hand, (16) is represented as

Variational Convexity of Functions in Banach Spaces

eλ ϕ(x) = eλv¯ ϕ(x − λv) ¯ + v, ¯ x −

259

λ λ v ¯ 2 = (eλv¯ ϕ ◦ θv¯ )(x) + v, ¯ 2, ¯ x − v 2 2

where θv¯ (x) := x − λv, ¯ x ∈ X , is an affine mapping. Thus the local convexity of ¯ Combining finally Theorem 4.6 eλv¯ ϕ around x reduces to that of eλ ϕ around x + λv. with the above equivalence, we arrive at the claimed result. 

References 1. Attouch, H.: Variational Convergence for Functions and Operators. Pitman, Boston (1984) 2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017) 3. Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303, 1–14 (2005) 4. Bernard, F., Thibault, L.: Prox-regularity of functions and sets in Banach spaces. Set-Valued Anal. 12, 25–47 (2005) 5. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005) 6. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations, vol. 2. Springer, New York (2011) 7. Browder, F.E.: Fixed point theory and nonlinear problems. Bull. Amer. Math. Soc. 9, 1–39 (1983) 8. Bynum, W.L.: Weak parallelogram laws for Banach spaces. Can. Math. Bull. 19, 269–275 (1976) 9. Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990) 10. Fabian, M., et al.: Functional Analysis and Infinite-Dimensional Geometry, 2nd edn. Springer, New York (2011) 11. Hare, W.L., Poliquin, R.A.: Prox-regularity and stability of the proximal mapping. J. Convex Anal. 14, 589–606 (2007) 12. Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (1993) 13. Khanh, P.D., Mordukhovich, B.S., Phat, V.T.: Variational convexity of functions and variational sufficiency in optimization. SIAM J. Optim. (2023). arXiv:2208.14399 14. Khanh, P.D., Yao, J.-C., Yen, N.D.: The Mordukhovich subdifferentials and directions of descent. J. Optim. Theory Appl. 172, 518–534 (2017) 15. Kurdila, A.J., Zabarankin, M.: Convex Functional Analysis. Birkhäser, Basel (2005) 16. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I. Basic Theory, II. Applications. Springer, Berlin (2006) 17. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham, Switzerland (2018) 18. Mordukhovich, B.S., Nam, N.M.: Convex Analysis and Beyond. I. Basic Theory. Springer, Cham, Switzerland (2022) 19. Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27, 170–191 (2002) 20. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, 2nd edn. Springer, Berlin (1993) 21. Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pacific J. Math. 33, 209–216 (1970) 22. Rockafellar, R.T.: Variational convexity and local monotonicity of subgradient mappings. Vietnam J. Math. 47, 547–561 (2019)

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23. Rockafellar, R.T.: Augmented Lagrangians and hidden convexity in sufficient conditions for local optimality. Math. Program. 192 (2022). https://doi.org/10.1007/s10107-022-01768-w 24. Rockafellar, R.T.: Convergence of augmented Lagrangian methods in extensions beyond nonlinear programming. Math. Program. (2022). https://doi.org/10.1007/s10107-022-01832-5 25. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) 26. Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)

Commutators on Power Series Spaces Over Non-Archimedean Fields ´ Wiesław Sliwa and Agnieszka Ziemkowska-Siwek

Abstract The power series spaces Ar (a) over non-Archimedean fields are the most known and important examples of non-Archimedean nuclear Fréchet spaces. In this paper we prove the following: (1) every operator on a stable power series space is a commutator; (2) every diagonal operator on a weakly stable power series space is a commutator; (3) the identity operator on any unstable power series space is not a commutator. Keywords Commutators · Non-Archimedean Köthe spaces · Power series spaces · Shift operators Mathematics Subject Classification: 46S10 · 47S10 · 47B47 · 46A35

1 Introduction Let E be a locally convex space (lcs). By an operator on E we mean a continuous linear map from E to itself. The algebra of all operators on E is denoted by L(E). The commutator of a pair of operators A and B on E is given by [A, B] := AB − B A. An operator T on E is said to be a commutator if T can be expressed in the form T = [A, B] for some operators A and B on E.

The research for the second author is supported by Ministry of Science and Higher Education in Poland: 0213/SIGR/2154. ´ W. Sliwa Institute of Mathematics, College of Natural Sciences, University of Rzeszów, Pigonia 1, 35-310 Rzeszów, Poland e-mail: [email protected]; [email protected] A. Ziemkowska-Siwek (B) Institute of Mathematics, Pozna´n University of Technology, ul. Piotrowo 3A, 60-965 Pozna´n, Poland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. M. Amigó et al. (eds.), Functional Analysis and Continuous Optimization, Springer Proceedings in Mathematics & Statistics 424, https://doi.org/10.1007/978-3-031-30014-1_12

261

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In 1947, Wintner ([14]) shown that any operator on an infinite dimensional complex Banach space E that is of the form αI + C, where α is a non-zero scalar, I is the identity operator on E and C is a compact operator on E, is not a commutator. In 1965, Brown and Pearcy ([3]) proved that an operator on the complex Hilbert space l2 is a commutator if and only if it is not of the form αI + C, where α is non-zero scalar, I is the identity operator on l2 and C is a compact operator on l2 . The same characterization of commutators was obtained in the case of the following complex Banach spaces: (1) l p , 1 < p < ∞, by Apostol in 1972 (see [1]); (2) c0 , by Apostol in 1973 (see [2]); (3) l1 , by Dosev in 2009 (see [6]). In 2010, Dosev and Johnson ([7]) proved that an operator on the Banach space l∞ is a commutator if and only if it is not of the form αI + S, where α is a non-zero scalar, I is the identity operator on l∞ and S is a strictly singular operator on l∞ . In 2015, Czy˙zak ([4]) proved that (1) every operator on a complex nuclear stable power series space is a commutator; (2) the identity operator I on a complex nuclear weakly stable power series space is a commutator. ´ In 2022, Sliwa ([13]) proved that every operator on an infinite-dimensional Banach space E over a non-Archimedean field K is a commutator if E is isomorphic to c0 (X, K) or l∞ (X, K), where c0 (X, K) is the Banach space of all functions f from X to K such that the set {x ∈ X : | f (x)| > ε} is finite for every ε > 0, and l∞ (X, K) is the Banach space of all bounded functions f from X to K (with the sup norm). Any infinite dimensional Banach space F of countable type (i.e. with a countable linearly dense subset) over K is isomorphic to c0 (N, K) (i.e. to the Banach space of all sequences (αn ) ⊂ K convergent to 0 with the sup-norm) ([11, Theorem 3.16]). Thus every operator on F is a commutator. By the Van der Put Theorem ([11]), for any infinite ultraregular compact space S the Banach space C(S, K) of all continuous functions from S to K, is isomorphic to c0 (X, K) for some infinite set X . Thus every operator on C(S, K) is a commutator. If the valuation of K is discrete (in particular, if K is locally compact) then any Banach space over K is isomorphic to c0 (X, K) for some set X . Thus every operator on an infinite-dimensional Banach space over a non-Archimedean field with a discrete valuation, is a commutator. In this paper we study commutators on power series spaces over non-Archimedean fields. By a non-Archimedean field we mean a non-trivially valued field K which is complete under the metric induced by the valuation | · | : K → [0, ∞) with the strong triangle inequality: |α + β| ≤ max{|α|, |β|} for all scalars α, β ∈ K. For any prime number p the field Q p of p-adic numbers with the p-adic valuation | · | p is non-Archimedean and locally compact. By Ostrovski’s theorem any complete non-trivially valued field that is not topologically isomorphic to the field of real numbers R nor to the field of complex numbers C is non-Archimedean (see [11]). From now on all linear spaces are over a non-Archimedean field K. By a norm on a linear space E (over K) we mean a function  ·  : E → [0, ∞) such that (1) x = 0 if and only if x = 0; (2) αx = |α|x for all α ∈ K, x ∈ E; (3) x + y ≤ max{x, y} for all x, y ∈ E (the strong triangle inequality). For fundamentals on locally convex spaces and normed spaces (over non-Archimedean fields) we refer to [8, 9, 11, 12].

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263

A sequence (e n ) in a lcs E is a Schauder basis in E if each x ∈ E can be written uniquely as x = ∞ n=1 αn en with (αn ) ⊂ K, and the coefficient functionals f n : E → ∈ N) are continuous. K, x → αn (n  A series ∞ x n=1 n in a Fréchet space E is convergent if and only if the sequence (xn ) is convergent to 0. Let E be a Fréchet space with a Schauder basis (en ). For any operator T on E there exists an infinite matrix (ti, j )i, j∈N of elements of ∞ ti, j ei for any j ∈ N; then K such that T e j = i=1 ⎛ ⎞ ∞ ∞ ∞    ⎝ Tx = ti, j x j ⎠ ei for x = x j e j ∈ E. i=1

j=1

j=1

An operator T on E is diagonal if its matrix is diagonal. By a Köthe space we mean an infinite-dimensional Fréchet space with a Schauder basis and with a continuous norm. An infinite matrix B = (bn,k ) of positive real numbers is a Köthe matrix if bn,k ≤ bn,k+1 for all k, n ∈ N. The space K (B) = {(αn ) ⊂ K : limn |αn |bn,k = 0 for every k ∈ N} with the base ( · k ) of norms, where (αn )k = max |αn |bn,k , k ∈ N, n

is a Köthe space. The sequence (e j ), where e j = (δ j,n ), is an unconditional Schauder basis in K (B). Any Köthe space is isomorphic to the space K (B) for some Köthe matrix B (see [5, Proposition 2.4]). A Köthe space K (B) is nuclear if for any i ∈ N there exists j ∈ N with j > i such that limn bn,i /bn, j = 0 ([5, 5, Proposition 3.5]). Let  be the family of all non-decreasing sequences a = (an ) of positive real numbers with limn an = ∞. Let a = (an ) ∈  and r ∈ {1, ∞}. The following Köthe space is the power series space: Ar (a) = K (B) with B = (bn,k ), bn,k = exp(rk an ), where rk = −1/k for all k ∈ N, if r = 1, and rk = k for all k ∈ N, if r = ∞. The power series space Ar (a) is of finite type if r = 1 and infinite type if r = ∞. Any power series space Ar (a) is nuclear. The power series space Ar (a) is: (1) stable, if supn (a2n /an ) < ∞; (2) weakly stable, if supn (an+1 /an ) < ∞; (3) unstable, if limn (an+1 /an ) = ∞. In this paper we prove the following: (1) Every operator on a stable power series space is a commutator (Theorem 2); (2) Every diagonal operator on a weakly stable power series space is a commutator (Theorem 3); (3) The identity operator on any unstable power series space is not a commutator (Theorem 5 and Corollary 6).

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2 Results We start with the following Lemma 1 Let Ar (a) be a weakly stable power series space (i.e. r ∈ {1, ∞} and a = (an ) ∈  with supn (an+1 /an ) < ∞). Then the shift maps ⎛ R : Ar (a) → Ar (a), R ⎝

∞ 

⎞ xjej⎠ =

j=1

⎛

and

L : Ar (a) → Ar (a), L ⎝

∞  j=1

∞ 

x j e j+1

j=1

⎞ xjej⎠ =

∞ 

x j e j−1

j=2

are well defined, linear and continuous. Proof Let C = supn (an+1 /an ). Let x = (x j ) ∈ Ar (a). Then |x j | exp(rs a j ) → j 0 for every s ∈ N. Let k ∈ N. Let p(k) ∈ N with p(k) > Ck. For j ∈ N we have x j e j+1 k = |x j | exp(rk a j+1 ) ≤ |x j | exp(r p(k) a j ) and x j e j−1 k = |x j | exp(rk a j−1 ) ≤ |x j | exp(r p(k) a j ). Indeed, consider two cases. Case 1: r = 1. Then rk = −1/k for k ∈ N. Hence rk a j+1 ≤ rk a j ≤ r p(k) a j and rk a j−1 ≤ (rk a j /C) ≤ r p(k) a j . Case 2: r = ∞. Then rk = k for k ∈ N. Hence rk a j+1 ≤ Crk a j ≤ r p(k) a j and rk a j−1 ≤ r p(k) a j . It follows that x j e j+1 k → j 0 and x j e j−1 k → j 0 for every k ∈ N. Thus x j e j+1 → j 0 and x j e j−1 → j 0 in Ar (a) for any x ∈ Ar (a), ∞ ∞   x j e j+1 and x j e j−1 are convergent in Ar (a) for any x ∈ Ar (a). so the series j=1

j=2

Therefore the maps R and L are well defined; clearly, these maps are linear. Moreover R and L are continuous, since   ∞     Rxk =  x e j j+1  = max x j e j+1 k ≤ max |x j | exp(r p(k) a j ) = x p(k)  j j  j=1  k

and

Commutators on Power Series Spaces Over Non-Archimedean Fields

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  ∞     L xk =  x j e j−1  x j e j−1 k ≤ max |x j | exp(r p(k) a j ) = x p(k) .  = max j≥2 j  j=2  k

 Now, we shall prove our main theorem, which indicates the class of nonArchimedean power series spaces on which every operator is a commutator. Theorem 2 Every operator T on a stable power series space Ar (a) is a commutator. Proof Let R and L be the shift operators on Ar (a), defined in Lemma1. Clearly, L R = I . Let c = supk (a2k /ak ). Then a j+n ≤ c(a j + an ) for all j, n ∈ N. ∞  ti, j ei for any j ∈ N. Let (ti, j )i, j∈N be the matrix of T . Then T e j = i=1

Let u ∈ N. By the continuity of T , there exist Mu > 0 and m u ≥ u such that T xu ≤ Mu xm u for all x ∈ Ar (a). For j ∈ N we have  ∞     max |ti, j | exp(ru ai ) =  ti, j ei  = T e j u ≤ Mu e j m u = Mu exp(rm u a j ). i   i=1

u

Thus for all i, j, m ∈ N with m ≥ m u we have (∗) |ti, j | ≤ Mu exp(rm a j − ru ai ). Let x = (xk ) ∈ Ar (a). For n ∈ N we have R TL x = R T n

n

n



∞ 

xk ek−n

=

k=n+1

∞  k=n+1

∞  i=1

xk R T ek−n = n

k=n+1

xk ti,k−n ei+n

∞ 

=

∞  i=1



∞ 

∞ 

xk R

k=n+1

xk ti,k−n ei+n

n

∞ 

ti,k−n ei

=

i=1

⎛ ⎞ ∞ ∞   ⎝ = x j+n ti, j ⎠ ei+n .

k=n+1

i=1

j=1

We will show that R n T L n x →n 0. Let p ∈ N. Let u, m, s ∈ N with u > 2cp, m > m u and s > 2cm. For some d > 0 and every n ∈ N we have   ∞     n n  R T L x p = max  ti, j x j+n  exp(r p ai+n ) ≤ max |ti, j ||x j+n | exp(r p ai+n ) ≤ i  i, j  j=1

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Mu max[|x j+n | exp(rs a j+n )] max[exp(−rs a j+n + r p ai+n + rm a j − ru ai )] ≤ j

i, j

Mu xs exp(−dan ) →n 0. Indeed, consider two cases. Case 1: r = 1. Then rk = −1/k for k ∈ N, −crs < −rm and d := rm − r p + ru > 0. Hence for all i, j, n ∈ N we have (−rs a j+n + r p ai+n + rm a j − ru ai ) ≤ (−rm (a j + an ) + r p ai+n + rm a j − ru ai ) = (−rm an + r p ai+n − ru ai ) ≤ (−rm + r p − ru )ai+n ≤ −dan . Case 2: r = ∞. Then rk = k for k ∈ N, r p c < ru and d := rs − ru − rm > 0. Hence for all i, j, n ∈ N we have (−rs a j+n + r p ai+n + rm a j − ru ai ) ≤ (−rs a j+n + ru (ai + an ) + rm a j − ru ai ) = (−rs a j+n + ru an + rm a j ) ≤ (−rs + ru + rm )a j+n ≤ −dan . It follows that R n T L n x p →n 0 for every p ∈ N, so R n T L n x →n 0. ∞  R n T L n x is convergent in Ar (a) for any x ∈ Ar (a). Thus the series n=0

By the Banach-Steinhaus theorem the linear map S : Ar (a) → Ar (a), Sx =

∞ 

Rn T L n x

n=0

is continuous. For all x ∈ Ar (a) we have (L(RS) − (RS)L)x = L R

∞ 

R TL x − R n

n

∞ 

n=0

=

∞  n=0

Rn T L n x −

∞  n=0

R n+1 T L n+1 x =

∞  n=0

n

R TL

n

Lx =

n=0

Rn T L n x −

∞ 

R n T L n x = R 0 T L 0 x = T x.

n=1

Hence T = L(RS) − (RS)L = [L , RS], so T is a commutator.



In the case of weakly stable power series spaces we get the following. Theorem 3 Every diagonal operator T on a weakly stable power series space Ar (a) is a commutator. Proof Let R and L be the shift operators on Ar (a), defined in Lemma 1. For some sequence (t j ) ⊂ K we have T e j = t j e j , j ∈ N.

Commutators on Power Series Spaces Over Non-Archimedean Fields

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Let p ∈ N. By the continuity of T there exist q ≥ p and C > 0 such that for every i ∈ N we have ||T ei || p ≤ C||ei ||q , so |ti | exp(r p ai ) ≤ C exp(rq ai ). Hence for every i ∈ N we have |ti | ≤ C exp(rq ai − r p ai ). Let x = (xm ) ∈ Ar (a). Then |xm | exp(rq am ) →m 0. For n ∈ N we have R n T L n x = ⎛ Rn T ⎝

∞ 

⎞

⎛

x j e j−n ⎠ = R n ⎝

j=n+1

∞ 

⎞ t j−n x j e j−n ⎠ =

j=n+1

∞ 

t j−n x j e j =

j=n+1

∞ 

ti xi+n ei+n .

i=1

We will show that R n T L n x →n 0. We have  ∞     n n ti xi+n ei+n  = max |ti ||xi+n | exp(r p ai+n ) ≤ R T L x p =  i   i=1

p

C max |xi+n | exp(r p ai+n ) exp(rq ai − r p ai ) ≤ i

C max |xi+n | exp(rq ai+n ) max exp(r p ai+n − rq ai+n + rq ai − r p ai ) = i

i

C max |xm | exp(rq am ) max exp[(ai − ai+n )(rq − r p )] ≤ C max |xm | exp(rq am ). m>n

i

m>n

It follows that R n T L n x p →n 0 for every p ∈ N, so R n T L n x →n 0. ∞  R n T L n x is convergent in Ar (a) for any x ∈ Ar (a). Hence, Therefore the series n=0

as in the proof of Theorem 2, we infer that the operator T is a commutator.



Hence we get the following Corollary 4 The identity map I on a weakly stable power series space Ar (a) is a commutator. Finally, we shall prove that the identity operator I on any unstable power series space Ar (a) is not a commutator. Theorem 5 Let E = Ar (a) be an unstable power series space and let V be an operator on E defined by the matrix (vi, j )i, j∈N . If V is a commutator, then vn,n →n 0. Proof Let S and T be operators on E with [S, T ] = V , defined by matrices (si, j )i, j∈N and (ti, j )i, j∈N , respectively. By the continuity of S and T there exist sequences (Ck ), (Dk ) ⊂ N and increasing functions p, q : N → N such that Sen k ≤ Ck en  p(k) and T en l ≤ Dl en q(l) for all n, k, l ∈ N. Hence |sm,n |bm,k ≤ Ck bn, p(k) and |tm,n |bm,l ≤ Dl bn,q(l) for all m, n, k, l ∈ N, where bn,k = exp(rk an ) for all n, k ∈ N. Thus for all m, n, k, l ∈ N we have: (∗) |sm,n | ≤ Ck exp(r p(k) an − rk am ) and ∞(∗∗) |tm,n | ≤ Dl exp(rq(l) an − rl am ). vl,n el ; on the other hand we have Let n ∈ N with n > 1. Then V en = l=1

´ W. Sliwa and A. Ziemkowska-Siwek

268

V en = [S, T ](en ) = ST en − T Sen = S

∞ 

tm,n em

−T

m=1 ∞ 

tm,n Sem −

m=1

∞ ∞   l=1

∞ 

sm,n T em =

m=1

tm,n sl,m el −

m=1

∞ ∞  

vn,n =

∞ 

tm,n sl,m el −

m=1

sm,n tl,m el =

sm,n em

=

m=1

m=1 l=1

l=1

Thus

∞  ∞ 

∞ 

∞  ∞ 

sm,n tl,m el =

m=1 l=1

∞ ∞   l=1

(tm,n sl,m − sm,n tl,m ) el .

m=1

(sn,m tm,n − sm,n tn,m ) = Wn1 − Wn2 + Wn3 − Wn4 ,

m=1

 n−1 ∞ 2 3 and where Wn1 = n−1 m=1 sn,m tm,n , Wn = m=1 sm,n tn,m , Wn = m=n+1 sn,m tm,n s t . Wn4 = ∞ m=n+1 m,n n,m To prove that vn,n →n 0 it is enough to show that Wni →n 0 for any i ∈ {1, 2, 3, 4}. Case i = 1. Let l = 1 and k = q(2). Then rq(l) < rk and rl < r p(k) . For 1 ≤ m < n, using (∗) and (∗∗), we get |sn,m ||tm,n | ≤ Ck exp(r p(k) am − rk an )Dl exp(rq(l) an − rl am ) = Ck Dl exp[(rq(l) − rk )an + (r p(k) − rl )am ] ≤ Ck Dl exp[(rq(l) − rk )an + (r p(k) − rl )an−1 ].

Hence |Wn1 | ≤ Ck Dl exp[(rq(l) − rk )an + (r p(k) − rl )an−1 ]. Since [(rq(l) − rk )an + (r p(k) − rl )an−1 ] = an [(rq(l) − rk ) + (r p(k) − rl )(an−1 /an )] →n −∞,

we infer that Wn1 →n 0. Case i = 2. Let k = 1 and l = p(2). Then r p(k) < rl and rk < rq(l) . For 1 ≤ m < n, using (∗) and (∗∗), we get |sm,n ||tn,m | ≤ Ck exp(r p(k) an − rk am )Dl exp(rq(l) am − rl an ) = Ck Dl exp[(r p(k) − rl )an + (rq(l) − rk )am ] ≤ Ck Dl exp[(r p(k) − rl )an + (rq(l) − rk )an−1 ].

Hence |Wn2 | ≤ Ck Dl exp[(r p(k) − rl )an + (rq(l) − rk )an−1 ]. Since [(r p(k) − rl )an + (rq(l) − rk )an−1 ] = an [(r p(k) − rl ) + (rq(l) − rk )(an−1 /an )] →n −∞, we infer that Wn2 →n 0. Case i = 3. Let k = 1 and l = p(2). Then rk < rq(l) and r p(k) < rl . For m > n, using (∗) and (∗∗), we get |sn,m ||tm,n | ≤ Ck exp(r p(k) am − rk an )Dl exp(rq(l) an − rl am ) =

Commutators on Power Series Spaces Over Non-Archimedean Fields

269

Ck Dl exp[(rq(l) − rk )an + (r p(k) − rl )am ] ≤ Ck Dl exp[(rq(l) − rk )an + (r p(k) − rl )an+1 ].

Hence |Wn3 | ≤ Ck Dl exp[(rq(l) − rk )an + (r p(k) − rl )an+1 ]. Since [(rq(l) − rk )an + (r p(k) − rl )an+1 ] = an [(rq(l) − rk ) + (r p(k) − rl )(an+1 /an )] →n −∞,

we infer that Wn3 →n 0. Case i = 4. Let l = 1 and k = q(2). Then rl < r p(k) and rq(l) < rk . For m > n, using (∗) and (∗∗), we get |sm,n ||tn,m | ≤ Ck exp(r p(k) an − rk am )Dl exp(rq(l) am − rl an ) = Ck Dl exp[(r p(k) − rl )an + (rq(l) − rk )am ] ≤ Ck Dl exp[(r p(k) − rl )an + (rq(l) − rk )an+1 ].

Hence |Wn4 | ≤ Ck Dl exp[(r p(k) − rl )an + (rq(l) − rk )an+1 ]. Since [(r p(k) − rl )an + (rq(l) − rk )an+1 ] = an [(r p(k) − rl ) + (rq(l) − rk )(an+1 /  an )] →n −∞, we infer that Wn4 →n 0. Corollary 6 The identity operator I on any unstable power series space Ar (a) is not a commutator. In the end we state the following open problems. Problem 1 Is every operator on a weakly stable power series space a commutator? Problem 2 Is a power series space Ar (a) [weakly] stable if every operator on Ar (a) is a commutator?

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