Basic Analysis V: Functional Analysis and Topology [1 ed.] 1138055131, 9781138055131

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Basic Analysis V

Basic Analysis V: Functional Analysis and Topology

The cephalopods are now fully trained and are now valued colleagues in Jim’s work.

James K. Peterson Department of Mathematical Sciences Clemson University

First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Peterson, James K. (James Kent), author. Title: Basic analysis I : functions of a real variable / James K. Peterson. Other titles: Functions of a real variable Description: First edition. | Boca Raton : CRC Press, 2020. | Includes bibliographical references and index. | Contents: Proving propositions -- Sequences of real numbers -- Bolzanoweierstrass results -Topological compactness -- Function limits -- Continuity -- Consequences of continuity of intervals -- Lower semicontinuous and convex functions -- Basic differentiability -- The properties of derivatives -Consequences of derivatives -- Exponential and logarithm functions -Extremal theory for one variable -- Differentiation in R² and R³ -Multivariable extremal theory -- Uniform continuity -- Cauchy sequences of real numbers -- Series of real numbers -- Series in general -Integration theory -- Existence of Reimann integral theories -- The fundamental theorem of calculus (FTOC) -- Convergence of sequences of functions -- Series of functions and power series -- Riemann integration : discontinuities and compositions -- Fourier series -- Application -Summary. Identifiers: LCCN 2019059882 | ISBN 9781138055025 (hardback) | ISBN 9781315166254 (ebook) Subjects: LCSH: Analytic functions--Textbooks. | Functions of real variables--Textbooks. | Mathematical analysis--Textbooks. Classification: LCC QA331 .P42 2020 | DDC 515/.823--dc23 LC record available at https://lccn.loc.gov/2019059882 ISBN: 9781138055131 (hbk) ISBN: 9780367768539 (pbk) ISBN: 9781315166155 (ebk) DOI: 10.1201/9781315166155 Publisher’s note: This book has been prepared from camera-ready copy provided by the author.

Dedication We dedicate this work to all of our students who have been learning these ideas of analysis through our courses. We have learned as much from them as we hope they have from us. We are firm believers that all our students are capable of excellence and that the only path to excellence is through discipline and study. We have always been proud of our students for doing so well on this journey. We hope these notes in turn make you proud of our efforts. Abstract This book introduces graduate students in mathematics to concepts from topology and functional analysis, both linear and nonlinear. We illustrate these ideas with a variety of real world applications, which can profitably use these concepts such as differential geometry and degree theory. We also show you how to try to find a proper abstract framework for interesting and difficult problems in the study of the immune system and models of cognition. We feel it is very important to realize that the hardest part of applying mathematical reasoning to a new problem domain is how you choose the underlying mathematical framework to use on the problem. Once that choice is made, we have many tools we can bring to bear which may be helpful. However, a different choice would let us do the analysis from a different perspective. We will discuss in detail these sorts of choices in our chosen applications. We feel this is a skill to have when your life’s work will involve quantitative modeling to gain insight into the real world. As usual, this book is designed for self-study. Acknowledgments I want to acknowledge the great debt I have to my wife, Pauli, for her patience in dealing with those vacant stares and my long hours spent in typing and thinking. You are the love of my life.

The cover for this book is an original painting by me which was done the summer of 2017 to serve as the cover for this book. It shows that the cephalopods have become fully trained in mathematics and have joined me as colleagues, as you can see from the problem we are working out on the blackboard in my office. I am fascinated by cephalopods and I look forward to having them as colleagues in the future!

Table of Contents I

Introduction

1

1

Introduction 1.1 Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6

II 2

3

4

III 5

Some Algebraic Topology Basic Metric Space Topology 2.1 Open Sets of Real Numbers . . . . . 2.2 Metric Space Theory . . . . . . . . 2.2.1 Open and Closed Sets . . . 2.3 Analysis Concepts in Metric Spaces 2.4 Some Deeper Metric Space Results . 2.5 Deeper Vector Space and Set Results

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Forms and Curves 3.1 When Is a 1-Form Exact? . . . . . . . . . . . . 3.2 Forms on More Complicated Sets . . . . . . . . 3.3 Angle Functions and Winding Numbers . . . . 3.4 A More General Definition of Winding Number 3.5 Homotopies . . . . . . . . . . . . . . . . . . .

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33 34 38 43 44 47

The Jordan Curve Theorem 4.1 Winding Numbers and Topology . . . . . . . . 4.2 Some Fundamental Results . . . . . . . . . . . 4.3 Some Applications . . . . . . . . . . . . . . . 4.3.1 The Fundamental Theorem of Algebra . 4.4 The Brouwer Fixed Point Theorem . . . . . . . 4.5 De Rham Groups and 1-Forms . . . . . . . . . 4.6 The Coboundary Map . . . . . . . . . . . . . . 4.7 The Inside and Outside of a Curve . . . . . . .

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Deeper Topological Ideas

77

Vector Spaces and Topology 5.1 Topologies and Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Topological Generalizations of Analysis Concepts . . . . . . . . . . . . . 5.1.2 Urysohn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

79 79 81 84

viii

TABLE OF CONTENTS 5.2 5.3 5.4

6

7

8

9

IV

Constructing Topologies from Simpler Sets . . . . . . . . Urysohn’s Metrization Theorem . . . . . . . . . . . . . . Topological Vector Spaces . . . . . . . . . . . . . . . . . 5.4.1 Separation Properties of Topological Vector Spaces

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88 90 94 98

Locally Convex Spaces and Seminorms 6.1 Additional Classifications of Topological Vector Spaces . . 6.1.1 Local Convexity Results . . . . . . . . . . . . . . 6.2 Metrization in a Topological Vector Space . . . . . . . . . 6.3 Constructing Topologies . . . . . . . . . . . . . . . . . . 6.4 Families of Seminorms . . . . . . . . . . . . . . . . . . . 6.5 Another Metrization Result . . . . . . . . . . . . . . . . . 6.6 A Topology for Test Functions . . . . . . . . . . . . . . . 6.6.1 The Test Functions as a Topological Vector Space . 6.6.2 Properties of the Topological Vector Space D( max{N3 , N4 , N5 }, by the triangle inequality, d2 (yp , zp ) < . Since  > 0 is arbitrary, we see yp = zp . Hence, fˆ is well-defined. It remains to show fˆ is uniformly continuous. Choose  > 0. Since f is uniformly continuous on A, there is a δ > 0, so that d1 (x, y) < δ

=⇒

d2 (f (x), f (y))
max{N1 , N2 }, we have δ 6 δ d1 (yn , y) < 6

d1 (xn , x)
max{N1 , N2 }, if d1 (x, y) < 6δ , d1 (xn , yn ) ≤

d1 (xn , x) + d1 (x, y) + d1 (yn , y)
0 with B(x, ) ⊆ GC . The exterior of a subset G is the set of all exterior points of G and is denoted by ext(G). With that said, it is straightforward to define boundary points of sets. Definition 2.3.13 Boundary of a Subset in a Metric Space The boundary of a subset G is the set of all elements x such that for every  > 0, B(x, ) ∩ int(G) 6= ∅ and B(x, ) ∩ ext(G) 6= ∅. The boundary of G is denoted by bnd(G) or ∂G. We can now characterize open and exterior sets better. First, boundary points: Theorem 2.3.9 Characterization of an Open Set in a Metric Space int(G) =

S

{O : O is open and O ⊆ G}

Proof 2.3.9 This one is for you.



Then the closure: Theorem 2.3.10 Characterization of the Closure of Set in a Metric Space G∗ =

S

{F : F is closed and G ⊆ F }

Proof 2.3.10 This one is for you.



And now the exterior: Theorem 2.3.11 Characterization of the Exterior Points of a Set in a Metric Space ext(G) = (G∗ )C Proof 2.3.11 This one is for you. Finally, the boundary. Theorem 2.3.12 Characterization of the Boundary of a Set in a Metric Space bnd(G) = G∗ \ int(G)



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BASIC ANALYSIS V: FUNCTIONAL ANALYSIS AND TOPOLOGY

Proof 2.3.12 This one is for you.



Homework Exercise 2.3.8 Prove Theorem 2.3.1 Exercise 2.3.9 Prove Theorem 2.3.2 Exercise 2.3.10 Prove Theorem 2.3.3 Exercise 2.3.11 Prove Theorem 2.3.4 Exercise 2.3.12 Prove Theorem 2.3.5 Exercise 2.3.13 Prove Theorem 2.3.6 Exercise 2.3.14 Prove Theorem 2.3.8 Exercise 2.3.15 Prove Theorem 2.3.9 Exercise 2.3.16 Prove Theorem 2.3.10 Exercise 2.3.17 Prove Theorem 2.3.11 Exercise 2.3.18 Prove Theorem 2.3.12

2.4

Some Deeper Metric Space Results

We know a subset of a metric space or a normed linear space is nowhere dense if its closure has an empty interior. A standard project in analysis is to construct Cantor sets and we can prove a Cantor set cannot contain any interval. Hence it is a nowhere dense subset of [0, 1]. We also prove that compact subsets of infinite dimensional normed linear spaces are nowhere dense. You can review these results in (Peterson (100) 2020). In (Peterson (100) 2020), we also discussed first and second category sets and for completeness and ease of discussion, let’s restate these discussions. Definition 2.4.1 First Category Sets of a Metric Space Let (X, d) be a metric space. We say X is of first category if there is a sequence (An ) of subsets of X which are all nowhere dense so that X = ∪∞ n=1 An . If X is not of first category, we say X is of second category. A most important result is then: Theorem 2.4.1 Baire Category Theorem Let (X, d) be a nonempty complete metric space. Then X is of second category. Proof 2.4.1 We assume X is of first category and derive a contradiction. This is a somewhat long proof so be patient! Since X is of first category, we can write X = ∪∞ n=1 Mn , where each Mn is of first category. (Step 1:) Since M1 is nowhere dense, M1 has no interior points; i.e. M1 contains no nonempty set. But X does contain nonempty open sets, trivially X itself, so we cannot have M1 = X. Hence (M1 )C = X \M1

BASIC METRIC SPACE TOPOLOGY

19

is nonempty and open as it is the complement of a closed set. Choose p1 ∈ (M1 )C . Since this set is open, there is a radius 1 > 0 so that B(p1 ; 1 ) ⊂ (M1 )C . We can assume 1 < 12 . (Step 2:) By an argument similar to that in Step 1, we know (M2 )C is both open and nonempty. Also M2 cannot contain an open set so B(p1 ; 21 ) ⊂ (M2 )C . This means (M2 )C ∩ B(p1 , 2 ) is a nonempty open set. Therefore there is p2 and associated radius 2 which we can choose so that 2 < 21 < 212 so that  ∈ (M2 )C ∩ B(p1 ; ), 2

p2

 B(p2 ; 2 ) ⊂ (M2 )C ∩ B(p1 ; ) 2

(Step 3:) Now continue by induction. We obtain a sequence (pn ) and a sequence (n ) satisfying n−1 ), 2 n−1 B(pn ; n ) ⊂ (Mn )C ∩ B(pn−1 ; ) 2 pn

∈

(Mn )C ∩ B(pn−1 ;

n
0. There is N so that B(pN ; 21N ). It follows that if m, n > N ,

1 2N

n−1 ) ⊂ B(pn−1 , n−1 ) 2

< η. We see then that if n > N , pn ∈ B(pN ; N ) ⊂

d(pn , pm ) ≤ d(pn , pN ) + d(pm , pN )
M . However, we also know n < 21n . This implies p ∈ B(pn ; n ) if 21n < ξ. Since ξ is arbitrary, we see p ∈ B(pn ; n ) ⊂ (Mn )C for all n. But that means p

∈

C ∞ ∩∞ n=1 (Mn ) = ∪n−1 Mn


C

by DeMorgan’s Laws. Thus, p ∈ X C where we assume X = ∪∞ n−1 Mn . But this is not possible. Thus our assumption X is of first category is wrong and we conclude X is of second category.  Comment 2.4.1 If (X, d) is a nonempty complete space, if we can write X = ∪∞ n=1 Mn , then at least one set Mp cannot be nowhere dense. So we know (Mp )C has a nonempty interior. Hence there is P ∈ Mp and a radius r > 0 so that B(p; r) ⊂ Mp . Also recall the important notion of compactness is expressible in terms of open sets. The two notions of compactness are sequential compactness and topological compactness which are defined as follows: Definition 2.4.2 Sequentially Compact Subsets of a Metric Space A subset G of a metric space is sequentially compact if every sequence in G has a convergent subsequence in G.

20

BASIC ANALYSIS V: FUNCTIONAL ANALYSIS AND TOPOLOGY

and Definition 2.4.3 Topologically Compact Subsets of a Metric Space A subset G of a metric space is topologically compact if every collection of open sets whose union contains G (this is called an open cover of G) has a finite subcollection whose union likewise contains G (i.e. a finite subcover of G). It then follows we can show Theorem 2.4.2 Sequentially Compact in a Normed Linear or Metric Space Implies Closed and Bounded Whether X is a normed linear space with norm k · k or a metric space with metric d, if S ⊂ X is sequentially compact, then it is closed and bounded. Proof 2.4.2 S is closed: Let (xn ) be a convergent sequence in S which is a subset of the normed linear space (X, k · k) . Then, there is x ∈ X so that xn → x in norm. Because S is sequentially compact, there is a subsequence (x1n ) which converges to y ∈ S. But the subsequential limit must be the same as the limit of a convergent sequence. Hence, y = x and we see S contains its limit points. So S is closed. If (xn ) is a convergent sequence in S which is a subset of the metric space (X, d). Then, there is x ∈ X so that d(xn , x) → 0. Because S is sequentially compact, there is a subsequence (x1n ) which converges to y ∈ S. But the subsequential limit must be the same as the limit of a convergent sequence. Hence, y = x and we see S contains its limit points. So S is closed. Note the argument for the norm and the metric are essentially the same. S is bounded: Let (xn ) be a convergent sequence in S which is a subset of the normed linear space (X, k · k) . Assume S is not bounded. Then for all n, there is xn ∈ S so that kxn k > n. But (xn ) is a sequence in S and so there is a subsequence (xnk ) which converges to y ∈ S. Then for  = 1 there is N so nk > N implies kxnk − xk < 1. The backwards triangle inequality for the norm gives nK > N implies kxnk k

≤

kxk + 1

and so kxnk k

≤ max{ max kxnk k, kxk + 1} 1≤nk ≤N

which says the subsequence is bounded in norm. But by construction, it is not. Hence our assumption the set is not bounded in norm is wrong. In the metric space setting, we have to decide what it means for a set to be bounded. A reasonable definition is that S is bounded if there is R > 0 so that d(x, y) < R for all x and y in S. Another way to look at this is to fix x0 in S and then boundedness of S means d(x, x0 ) < R for all x ∈ S. Let’s assume S is not bounded. Then for all n, there is xn so that d(xn , x0 ) > n. But (xn ) ⊂ S and so it has a convergence subsequence (xnk ) which converges to x ∈ S. Thus for  = 1, there is N so that nK > N implies d(xnk , x) < 1. Now apply the backwards triangle inequality to find d(xnk , x0 )
0, A can be covered by a finite number of subsets of X whose diameters are all less than epsilon.  That is, given  > 0, there is a finite collection {S1 , . . . , Sp } so that A ⊂ ∪pi=1 Si with diam(Si ) < . This finite collection of points is also called an -net.

22

BASIC ANALYSIS V: FUNCTIONAL ANALYSIS AND TOPOLOGY

We now prove a useful result called Lebesgue’s Lemma which we use in our development of the Jacobian Curve Theorem. Lemma 2.4.4 Lebesgue’s Lemma in a Metric Space Let Gα be an open cover of the compact set S in the metric space (X, d). Then there is an  > 0 so that for all x ∈ S, B(x, ) ⊂ Gα for some α. Proof 2.4.4 Let Gα be an open cover of S. Assume this is not true. Then for all n, there is a point xn ∈ S with B(xn , n1 ) 6⊂ Gα for all indices α. Since S is sequentially compact and (xn ) ⊂ S, there is a subsequence (x1n ) and a point x ∈ S with x1n → x. Since (Gα ) is an open cover of S, there is an index β so that x ∈ Gβ . Hence, x is an interior point of Gβ . So, there is a δ > 0 so that B(x, δ) ⊂ Gβ . But also, there is an N so that d(x1n , x) < 2δ for all n1 > N . Thus, if y ∈ B(x1n , n11 ), d(y, x) ≤ d(y, x1n ) + d(x1n , x) < We see if we choose M so that

1 n1

1 δ + n1 2

< 2δ , if n1 > max(N, M ), we have

d(y, x)
0 so that B(x, ) ⊂ Gα for some α.  We can then prove a very nice equivalence theorem for these concepts which is not tied to 0 so that B(x0 , δ) ⊂ Oβ . Then there is N with 1/N < δ/2 so that d(x0 , xn ) < δ/2 if n > N . This tells us C nN ⊂ B(xN , 1/N ) ⊂ B(xN , δ/2) ⊂ B(x, δ) ⊂ Oβ But this set cannot be covered by a finite number of Oα so this is a contradiction. Hence, A must be topologically compact.  The following theorems state the fundamental properties of open and closed sets within a given metric space X: first, the open sets: Theorem 2.4.6 Properties of Open Sets in a Metric Space For any metric space X: 1. The empty set ∅ is an open set. 2. The space S is an open set. 3. Any union of open sets is itself an open set. 4. Any finite intersection of open sets is itself an open set.

Proof 2.4.6 We leave the proof of these results to you. And now the closed sets: Theorem 2.4.7 Properties of Closed Sets in a Metric Space



24

BASIC ANALYSIS V: FUNCTIONAL ANALYSIS AND TOPOLOGY

For any metric space X: 1. The empty set ∅ is a closed set. 2. The space S is a closed set. 3. Any intersection of closed sets is itself a closed set. 4. Any finite union of closed sets is itself a closed set.

Proof 2.4.7 We leave these results to you.



It is apparent that Theorem 2.4.7 is obtained by directly applying De Morgan’s Laws to Theorem 2.4.6 and vice versa. Also, we note that when we say any union of open sets and any intersection of closed sets we really do mean any whether the union or intersection is finite, countably infinite, or even uncountably infinite. Likewise, the restriction of open sets to finite intersections and closed sets to finite unions is truly necessary. It is not correct in general that an intersection of open sets is open or that a union of closed sets is closed. Homework Exercise 2.4.1 Prove [a, b] is totally bounded in < by finding the -net for a given choice of . Exercise 2.4.2 Prove [a, b] × [c, d] is totally bounded in 0 0, x≤0

This is C ∞ with limx→0+ f0 (x) = 0 and limx→∞ f0 (x) = 1. A messy induction proof then shows f0 is C ∞ with all the derivative zero at x = 0. This is easily generalized to  − 1 e x−a , x > a fa (x) = 0, x≤a which is C ∞ with all derivatives and the original function zero at x = a. In fact, if we define  1 1 1 e− x e− 1−x = e− x(1−x) , 0 < x < 1 g(x) = f0 (x)f0 (1 − x) = 0, x < 0 or x > 1 We see g is C ∞ with all derivatives and the function itself zero at x = 0 and x = 1. We can adapt this to the interval [a, b] to get     x−a x−a g(x) = f0 f0 1 − b−a b−a     x−a b−x = f0 f0 b−a b−a ( (b−a)2 e− (x−a)(b−x) , a < x < b = 0, x < a or x > b which has support [a, b] and all derivatives and the original function are zero at x = a and x = b. We can then use this on the rectangle R = (a1 , b1 ) × (a2 , b2 ) × . . . × (an , bn ) to define the function h on 0 so that if the diameter of a set W in U is less than epsilon, there is a set Ui with W ⊂ Ui . Let A ⊂ U be compact. Then A is contained in a finite union of open sets Ui . Letting N be the largest such index, we see A ⊂ ∪N i=1 Ui . Now any rectangle R ⊂ A with diameter less than  is contained in some Ui . Subdivide the compact set A using a partition Π on each axis so that kΠk < 2 . The partition then gives a collection of rectangles (Rα ) with diam(Rα ) < . Thus, each Rα ⊂ Uiα for some index iα . Each rectangle will then have a C ∞ bump function hα with supp(hα ) = Rα and hα = 0 on (Rα )C . Define these functions: X σ1 = hα =⇒ supp(σ1 ) ⊂ U1 α:Rα ⊂U1

σ2

=

X

hα =⇒ supp(σ2 ) ⊂ U2

α:Rα ⊂U2

.. . σi

=

X α:Rα ⊂Ui

hα =⇒ supp(σi ) ⊂ Ui

BASIC METRIC SPACE TOPOLOGY

27

Since we only have to consider up to UN , this process terminates at σN . We set σj = 0 when j > N . By Theorem 2.4.8, we know we can write U = ∪∞ i=1 Ki where each Ki is compact and Ki ⊂ int(Ki+1 ) for all i ≥ 1. Set K0 = ∅ and K−1 = ∅. Define A1 = K1 \ int(K0 ) = K1 ,

W1 = int(K2 ) \ K−1 = int(K2 )

A2 = K2 \ int(K1 ),

W2 = int(K3 ) \ K0 = int(K3 )

A3 = K3 \ int(K2 ),

W3 = int(K4 ) \ K1 = int(K3 ) .. .

Aj = Kj \ int(Kj−1 ),

Wj = int(Kj+1 ) \ Kj−2 = int(K3 )

For example, look at x ∈ A3 . Then x ∈ K3 ⊂ int(K4 ) ⊂ K4 and x ∈ (int(K2 ))C . Our chain conditions then tell us (int(K2 ))C ⊂ K1C . Hence, x ∈ K4 \ K1 = W3 . Similar arguments hold for all the cases: so we know Ai ⊂ Wi for all i. It is straightforward to see U = ∪i Ki = ∪i Ai . Thus, we have A1

⊂ W1 ,

A1 compact

A2

⊂ W2 ,

A2 compact

A3

⊂ W3 , .. .

A3 compact

Aj

⊂ Wj ,

Aj compact

So all Wi are open and all Ai are compact. Also, from the conditions, we see Aj ∩ Aj−1 = ∅ so this is a sequence of disjoint sets. Since Aj ⊂ Wj , we see Aj ⊂ ∪i Ui ∩ Wj and so (Ui ∩ Wj ) is an open cover of Aj . Now apply the argument we used for the compact set A earlier to each Aj . We obtain a set (σij ) with supp(σij ) ⊂ Ui ∩ Wj ⊂ Aj . For each index j, only a finite number of σij (x) P > 0 for any x ∈ Aj nj as we showed in our earlier construction. Let this number be nj . Define ψj = i=1 σij . It follows supp(ψj ) = Aj . P∞ Define ψ(x) = j=1 ψj (x). Given x ∈ U , x ∈ Aj for only one index j0 . By the finite union condition, there are at most a finite number of Ui with x ∈ Ui . Now if there were an infinite number of indices such that σik ,jk (x) > 0, then x ∈ Uik ∩ Wjk . This says x is in more than a finite number of sets Ui . This is not possible because of the finite union condition. Hence, at most a finite number of σij ’s can be positive at any x. Hence, we can write ψ(x)

nk X

=

k=1

σik ,jk (x) =

nk X

ψjk (x)

k=1

Hence, ψ(x) is finite, the infinite sum is well-defined and ψ > 0 on U . Finally, define fi = is positive on U and at any x ∈ U , we have ∞ X i=1

∞

fi (x)

=

ψj ψ

which

n

k X 1 X 1 ψj (x) = Pnk ψjk (x) = 1 ψ(x) j=1 k=1 ψjk (x)

k=1



28

BASIC ANALYSIS V: FUNCTIONAL ANALYSIS AND TOPOLOGY

Homework Exercise 2.4.5 In the proof of the Cantor Intersection Theorem, prove that since the Cauchy sequence (xn ) obtained in the argument converges to x ∈ X, x ∈ ∩n Fn . Exercise 2.4.6 Go back and revisit the original proofs of the Bolzano - Weierstrass Theorem for a bounded infinite sequence and this time use the Cantor Intersection Theorem to find show the subsequential limit is in the set. Exercise 2.4.7 Pick any open bounded set in