Summer School on Topological Vector Spaces (Lecture Notes in Mathematics, 331) 3540063676, 9783540063674

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zurich

331 Summer School on Topological Vector Spaces

Edited by Lucien Waelbroeck Universite Libre de Bruxelles, Bruxelles/Belgique

Springer-Verlag Berlin· Heidelberg' NewYork 1973

AMS Subject Classifications (1970): 46-02, 46 A xx

ISBN 3-540-06367-6 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-06367-6 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer Verlag Berlin' Heidelberg 1973. Library of Congress Catalog Card Number 73-83244. Printed in Germany. Offsetdruck: Julius Beltz, Hernsbach/Bergstr.

PREFACE This volume contains lecture notes for five of the six series of lectures at the Summer School on Topological Vector Spaces, held at the Universite Libre de Bruxelles in September 1972, and a summary of the four invited Seminar talks. The missing series of lectures is that of L.TZAFRIRI, on Special Banach Spaces.

A separate issue, probably in this Lecture

Notes series, co-authored by L.TZAFRIRI and J.LINDENSTRAUSS, will be an expanded summar} of TZAFRIRI's talks. Let all those who helped make this Summer School a success find here an expression of my gratitude, contributors, participants. the secretaries of my Mathematics Department, and foremost the NATO Science Committee who run a very effective Summer School program and financed this specific meeting.

Lucien WAELBPOECK

TABLE OF CONTENTS

Lucien WAELBROECK

"Topological vector spaces"

"locally convex spaces" Henri HOGBE-NLEND Joseph WLOKA

"Gelfand triplets and spectral theory"

Henri BUCHWALTER Ernst BINI Marc DE WILDE

"Techniques de bornologie en des espaces vectoriels topologiques"

"Fonctions continues et mesures sur un espace compl etement ier"

"Convergence structures on S(X)" "Various types of barrelledness and increasing sequences of balanced and convex sets in locall} convex spaces"

.

1

41

84

163

183 203

211

David J.H.GARLING : "Lattice bounding mappings"

218

Philippe TURPIN: "Linear operators between Orlicl spaces"

222

CONTRIBUTORS Lucien WAELBROECK.

Universite Libre de Bruxelles

1050 Bruxelles, Belgique. John HORVATH.

University of Maryland

College Park, Maryland 20742. U.S.A. Henri HOGBE-NLEND.

Universite de Bordeaux

33405 Talence, France. Joseph WLOKA.

Universitat Kiel

23 Kiel, Deutschland. Henri BUCHWALTER.

Universite de Lyon I

69621 Villeurbanne, France. Ernst BINZ.

Universitat Mannheim

68 Mannheim, Germany. Marc DE WILDE.

Universite de Liege

4000 Liege, Belgique. David G.H.GARLING.

Cambridge University

Cambridge CB3 9DA, England. Philippe TURPIN.

Universite de Paris-Sud

Centre d'Orsay. 91405 Orsay, France.

- 1 -

TOFOLOGICAL VECTOR SFACES I . WAELB ROECK 1. Generalities

A topological vector space

1.1.

(E, 'b)

is a k-vector space, with a topol09J tions continuous.

on a topological field

k

making the algebraic opera-

What I intend to saJ applies to real and to com-

plex vector spaces.

But I have acquired poor habits, and will only

consider in these notes complex spaces, henceforth

k

=t

with its

usual t.opol cqy . We wish the addition map cation

. : t

x

E

+ :

E

x

E

E to be continuous.

is a topological group.

The topol09J

V of neighbourhoods of the origin.

0 And

E and scalar multipliA topological vector space is determined bJ the filter

v

is the filter of neigh-

bourhoods of the origin for a vector space topologJ when a.

V U EV ] V E \/: U :J V + V

b.

The filter

c.

The el ements of If

J

has a balanced basis are absorbing.

A proof of the fact that the filters with properties a, b. c are exactlJ

the neighbourhood filters for vector space topologies is con-

tained in all standard texts on topological vector spaces (cf. e.g. [71, paragraph 15 or [19 J chapter I, paragraph 1).

The definition of

a balanced set and of an absorbing set must be given however. ACE lsi an

1. £

is balanced when

sa E A follows from

On the other hand, ACE

> 0 can be found such that

is absorbing when for every se E A follows from

observe that the set of absorbing subsets of balanced basis.

a E A, sEt,

E

lsi


O}

is the

basis of the neighbourhood filter for a local1} convex topology weaker than the given one.

The locally convex (semi-normed) space ob-

tained in this way is called

EV; the elementary properties of locally convex spaces are obtained by exhausting in some way or; by means of these semi-normed topologies. When

V is balanced. absorbing. but not convex. {sV

is not in general a neighbourhood basis.

I lsi>

O}

To obtain a neighbourhood

basis. we have to start out from a balanced neighbourhood

V. in a

topological vector space. and define

Vk inductively in such a way is a balanced. absorbing neighbourhood for all k. and

that

Vk Vk_1 :? Vk + Vk' We can define an "elementary neighbourhood ahain" as a sequence (VI' V2' ... ) with each Vk balanced, absorbing, and quite a bit of the baby Vk :? Vk+ 1 + Vk+ 1' The reader will

work that can be done in the locally convex case through the consideration of the space

EV' will be repeated here, but replacing absolutely convex absorbing V by elementary neighbourhood chains. Norms, semi-norms, and systems of norms and semi-norms define

1.3.

locally convex topologies.

To obtain general vector space topologies.

we need more general objects. ft,n

If

3-semi-norm is a mapping

a.

'..' (x + y) .;;;; v(x) + ') (j)

b.

vOx) .;;;; \' (x)

c.

V(AX) v

These are the i-norms and l-semi-norms.

for

for all

,

v

for

x,

scalar,

x

and

: E

such tn at

F+

y E E;

1>..\ .;;;; 1

x scalar.

does not vanish off the origin. then

and A v

x

E

E;

O.

is an ;1-norm.

It is clear that an 1-norm, an 1-semi-norm, or a system of 3-norms and !-semi-norms will define a vector space topology in

waj

- 3 -

quite analogous to the locally convex case.

If

v

is an g-semi-norm

and

then

VI" ... , Vk, ... is an "elementary neighbourhood system". Conversely, everJ vector space topology can be defined by a sys-

tem of :I-norms or 1-semi -norms.

What we need to to is associate an

1-semi -norm to each of the elementary neighbourhood chains. Let thus let

VI"

s,O",s",1

k, q

integers);

be an elementarJ neighbourhood chain; be a dyadic rational (i.e. a number 2 -k q with express

s

in the dJadic scale s

where the

kp

=

2

-k

p

are integers, different from each other, and let

Ws = E when s ;;;. l. A little thought will show that W c W s ' , and s' when s s that Ws + W C W . A hint maJ be useful to help the reader s2 - sl+ s2 1 to think. Complete this definition, putting

"

The ordering of dJadic rationals is, of course, the phic ordering of their sJstems of digits in the dJadic scale. assume that S

I

for

s < s'.

This implies that the first

coincide, but that for some s'

is

j ,

the digit for

This gives us a term

Vj

of s

is

We sand

0, that

in the expression of

and no such term in the expression of

Ws I, At most the furtherdigits

for

s

s'

unitJ.

Ws' would all vanish, while those for

This would give us something like

expression of Vj

l.

W s.

Vj+ 1 + Vj+ 1

would all be equal to

Vj+ 1 + ... + Vk in the But an inductive application of the relation

shows that V. ::> V. 1 + •.. + Vk J J+

- 4 -

W + W C W i 5 not very dif51 52 - 51+5 2 What one really must do is carrJ out the addition s1 + 52

The idea of the proof that ferent.

in the dyadic scale. look out for what happens at each carrj over. and. when considering the 5ets

Ws

remember that

V. + V.J c V.J- l' J

We then define

It is clear that

v

is an ]-semi-norm whose kernel is exactlJ the

intersection of the neighbourhoods bj the sequence of sets

Vk

Vk. Also. filter generated is the same as that generated bJ the

sets {x

We

I

v(x)

e;}

have shown that vectorwpologies could be generated bJ J-norms

and 1-semi-norms. The reciprocitj between absolutelJ convex absorbing sets and usual norms and semi-norms is alreadj not perfect in the locallj convex case.

It is even worse here.

But it is good enough if we are

concerned with neighbourhood filters. rather than individual neighbourhoods. The fact that a Hausdorff vector space topologj is metrizable if and onlJ if the origin has a fundamental sequence of neighbourhoods is clear. (and well known).

A metrizable vector space topologj can

also be defined bj a single }-norm. 1.4.

A subset

uniformlJ for that an

E

B of a topological vector space is bounded if s

scalar. s

> 0 can be

origin in such a waJ that 5ubsets of E.

O. and

x

ranging over

B.

to everj neighbourhood sB C U when

!sl

E.

sx

This means U of the

The set of bounded

E is clearlj a vector space boundedness (bornologj) on

It i5 the von Neumann boundedness of

E.

Another bounded structure can be defined canonicallj on a topo-

0

- 5 -

logical vector space. the additive boundedness.

B is additivelj

bounded iff one can associate a positive integer bourhood

k

to everj neigh-

U of the origin in such a waj that BCU+ ... +U

(k

terms)

Additivelj bounded subsets of locallj convex spaces are bounded. in general. U + ... + U is much larger than

But

kU. we must expect the

additive boundedness to be much grosser than the usual. von Neumann boundedness. We shall see that a set

B is additivelj bounded iff everj

continuous 3-semi-norm is bounded on

B.

It is clear that continuous

3-semi-norms are bounded on all additivelj bounded sets. assume that

B is not additivelj bounded. let

of the origin such that Let

v

OJ 1

U be a neighbourhood

U + •.. + U for anj number

be an 3-semi-norm such that

v(x) > 1 for

k of

x, U.


0 and

a real number

M exist such that

p{Mt)

(I +

for all

- 9 -

If the condition is satisfied, the ball of radius the ball of radius

(1

absorbs

E).a, hence all balls of finite radius, and

+

better, the ball of radius radius.

a

a

is absorbed by all balls of non zero

As a matter of fact, if

f o ( I f I) dm we see that

" (1 + c ) a

f o ( I f II M) dm

" a

and this proves the result. This sufficient condition cannot be necessary in all cases. If

m is a finite measure, Lo

depends only on the behaviour of

in the neighbourhood of infinitj.

For

Lp

p

to be locallj bounded.

it is therefore sufficient that 1i min f t-- P(M t ) I p( t) > 1 fol" some

M E IR+.

Similarly, when we consider behaviour of

p

as

t

0, hence

.I'.

p

will be locally bounded if

p(Mt)/p(t) > 1

lim 2.5.

lp' the space depends only on the

On the other hand, assume that

that

m is not purelj atomic and

lim inf t __ p(Mt)/p(t) = 1 for all Lp (n,a.. m) is not 1oca 11j bounded. Since

1i m inf p(Mt)/p(t)

find a sequence M.

We choose

tn s

>

00

0

'=

1

such that

and let

an

M.

for all

We shall see that it is possible to

M,

p(Mtn)/p(t n) 2r/p(tn)·

1

for all

We also choose

n

En c 0

with

mEn = an' such a set exists if n is large enough. We next define a function f n = t n on En' f n = 0 off En' The function f n belongs to the closed ba 11 of radius 2e::. This sequence is however not absorbed bj the ba11 of radius

E.

and

Jus t

- 10 -

cons i der

since

p(tn)mE n

= 2£

and

p(tn)/p(tn/M)

1.

This proof can be adapted in cases where

m is pure1j atomic,

and has atoms of sufficient1j large measure, saj the atom of order Also, if p(Mt)/p(t) 1 as t has a measure larger than a- k for all

k 00

M, it is not difficult to change the above proof and show

that

Lp is not 10ca11j bounded as soon as ri1j small measure.

m has atoms of arbitra-

I have not carried out the computations, but I am convinced that a 10ca11j bounded would be

Lp

could be constructed, where the measure space

IN, the atom

k

having the measure

11k!.

and where

lim inf t 100 p(Mt)/p(t) = 1 for all M. If I maj state a stronger conjecture: I am convinced that somebodj has a1readj carried out these computations. 2.6.

Assume now that

that

n

o
1

lim lp(l)

k

is 10ca11j bounded iff some

k

(a)

exists with

p(kt)/p(t) > 1

lim

(b)

lp(lR) is 10ca11j bounded iff conditions (a) and (b) both hold. 2.8.

We must fina11j speak of the additive boundedness of the Or1icz

spaces.

We shall see that this is equivalent to the metric bounded-

ness in the three standard cases. i.e. when n = I. rl = F • or

n =

each of theses spaces being equiped with the standard measure.

When.

rl

=

}l •

we shall assume that

p(t)

co

as

t

"".

This does not af-

fect the topo10gj or the additive boundedness. but it does affect the metric boundedness. A set is metrica11j bounded in a metric space if it has finite diameter.

We consider on an Or1icz space the distance associated to

the :I­norm vp(f) = J p(lfl)dm It is clear that an additive1j bounded set is metrica11j bounded.

- 12 -

Assume conversely that where

is a diffuse measure space.

Q

where

k>Mh can split and

f

vp(f i)

M E lR+

= f1 + = M/k
0

Let

o

k

and

E

}l •

v (f) < M when f E B. We o + f k where the f i "sit" on disjoint sets This shows that B C :r1k U. i f U is the open is such that

i . e.

is additively bounded.

B

lp. assume that

metrically bounded in

L (Q)

is metrically bounded in

B

pet)

as

00

t

00.

Let

B be

lp. B is contained in a ball of radius

with center at the origin.

We can split each

a E B as

a

M

= a1

+ a2

aI' a 2 have disjoint supports, where the absolute values of the components of a 1 are larger than n if pen) = E/2, while the absolute values of the components of a 2 are at most n. The set of

where

al

under consideration is bounded in

supported by a set with

k

loo' and each of its elements is

elements. where

k > 2M/e:.

It is not dif-

ficult to show that such a set of elements is additively bounded in ,(

p

. We must still show that

a2

ranges over an additively bounded

Of course. v p(a 2) < M, the components of a 2 are less than n with pen) £/2. We can split each a 2 under consideration as a2 = bo + b I + ... + br where the bk "sit" set. when

a

ranges over

B.

on disjoint sets. and where

£/2 < v p (b.) < 1

These inequalities imply that

r

under considertation belongs to £

E,

= 1•...• r.

when

M/2£ < k, hence each of the U when

U is a ball of radius

and center at the origin. We observe that the additive and the metric boundedness of

do not coincide any more when of

a2

lp

p

is a bounded function.

is metrically bounded as soon as an integer

such that each

b E B sits on a set with

k

k

elements.

E

lp

A subset exists The metric

boundedness is not separated. but the additive boundedness is separated.

B

- 13 3. Variants on the notion of an Orlicz space

3.1.

The Orlicz spaces are the first examples of non locallj convex

spaces that one encounters.

It is well known, but it must however

be said explicitlj, that the space of measurable functions on a finite measure space, with convergence in measure, is a special Orlicz space.

Just take, for

t

E

IR+, t

p(t}

l+t

More general spaces of Orlicz tjpe would be obtained in the fol-

(n,o., m)

lowing waj.

p: II

be a mappi ng that

p(x, t}

each

x.

would be a measure space, but now IR+

x

-+

IR+ ' whi ch woul d be measurable, and such

is subadditive in

The space

L

p

would

p

(lI, 0., m)

t

and tends to zero as

t

-+

for

0

would be the space of functions

f

such that v o (f)

= f o I x , I f I} dm
) 6l E which approximate u uI'd form 1jon the i r dom a ins. Tietze's theorem allows us to extend

• 30 -

elements of

C(X i) A E to

C(X) A E.

vi' and call these extensions again Let let

We consider such extensions of vi.

U be an open balanced neighbourhood of the origin in

V be open. balanced. such that

above gives us functions when

x E Xi.

then

Ai

¢

V + V.

The construction

vi E C(X) i E such that

u(x)· vi(x) E V

Let

is open. Ai

a function

U

E.

Xi.

Urisohn's theorem shows the existence of

with compact support in

A2. The function v

AI' such that

- ¢

has com·

pact support in

=¢

u(x) - v(x) E U for all

vI

(1 . ¢)v 2 x E X. +

belongs to

C(X) A E. and

The above remarks do not tend to make the existence of a compact set which does not have the densitj propertj un1ike1j. show that such a space 8.2.

Thej on1j

X is difficult to construct.

The density problem is related to the approximate extension pro·

b1em. or when Let

E is a metrizab1e space. to the extension problem.

Y be compact. let

X he a closed subset of

E be a topological vector space.

(X, Y, E)

tension property if restriction maps C(X. E). (X. Y, E)

Also. (X. V)

r(Y, E)

Y, and let

has the approximate exon a dense suhset of

has the approximate extension property if

has the approximate extension propertj for all

has the approximate extension if tension propertj for all (X, I. E) C(I. E)

onto

couples

XC I

(X. Y, E)

E.

And

E

has the approximate ex-

X C Y.

has the extension propertj if restriction maps C(X. E).

The definition of the extension propertj for

of compact spaces and for topological vector spaces

E is clear and will be left to the reader. (X. I. E)

has the approximate extension propertj if

(X, E)

has

- 31 -

the densitj propertj.

Converselj. if

(i. E)

extension propertj. and if (X. E)

has the rlensitj propertj. then

(X. E)

u E C(X. E).

has the densitj propertj. and that

We can find

v.

Then

u

XCi.

v E C(X) i E approximating

is

u.

w E C(i) A E exten-

Tietze's theorem proves the existence of some ding

has the approximate

has the densitj propertj.

Assume that Let

(X. i. E)

extendable. and

(X. i. E)

has

the approximate extension propertj.

(X. I. E)

Converselj. assume that sion propertj and that u E C(X. E).

has the densitj propertj.

v E C(i. E)

We can find

which extends

w E C(I) A E which approximates

telj. then tion of

(I. E)

has the approximate

w to

X belongs to

C(X) A E and

v

on

.

let

u

I.

approximaThe restric-

approximates

u.

This result relates the approximate extension problem verj directly to the density problem. probability measures on

We let

0

is said to be: absorbing if for

1/)

A locally of

which satisfy:

and

W:> V,

then

"'1 E

10 ,

(2) every VE 10 is absorbing, (3) any finite intersection of sets belonging to 10

contains a

balanced, convex set belonging to 10 (4) i f V E 10

A > 0,

and

AV E 10 .

then

A vector space equipped with a locally convex structure is called a locally convex space.

Any V E 10 contains the origin

10 belongs to 10

section of sets belonging to

A -J 0, then

E and

F

(by 3 and 1); i f V E 10 and

be two locally convex spaces whose structures are

defined by the collections 10

and 110,

respectively.

is a morphism of locally convex spaces if f: E

belongs to f E

(by 3); any finite inter-

AV E 10 (by 3, 4 and 1). Let

morphisms

0

;c (E,F),

F

form a vector space

(E,E); if

A linear map

(VI) E 10 for all

;;c, (E,F).

f: E-+F

VI E tv).

The

The identity map

are three locally convex spaces, and

E,F,G

g E. X. (F,G),

f

-1

gaf E :t (E,G): the locally convex spaces form

then

a category. Let ""t:

1.3. (x,y)

x + y

from

E

>< E

be a topology on into

E

and

E

( A, x )

such that the maps

.!I x from

are continuous and assume that

0

which consists of convex sets.

Then the collection 10

of

0

IK x E

into

E

has a fundamental system of neighborhoods

satisfies conditions (1) - (4).

of all neighborhoods

Conversely, if a locally convex

- 42 structure is defined by a collection

x + V,

then the sets

will be the collection of all neighborhoods of the point which the maps

x

where

for a topology for

are continuous and each point

possesses a fundamental system of convex neighborhoods. the oontinuous linear :maps.

Themorphisms are then

This method of introducing locally convex spaces

(i.e., locally convex topological vector spaces) avoids introducing preliminary topological concepts [75].

1.4.

Let 1!J.-

the vector space and

be a collection of absorbing, balanced, convex subsets of

E.

10

Then the collection

of all sets

contains a finite intersection of sets belonging to

V

A W, A> 0,

convex structure on tions

, or equivalently

satisfies (1) - (4) and so defines a locally

WE!{f,

E,

0

V which contain a finite intersection of sets of

the collection of all sets the form

A>

where

V,

said to be generated my

Two different collec-

can generate the same locally convex structure.

all balanced, convex subsets belonging to

The collection of

generate the locally convex

structure defined by 1) .

A semi­norm on

E

is a map

positive real numbers which satisfies

IAI

p(x )

for all

x,y E E,

/\ Ell 0,

of

1

J

balanced, convex the closed se.mi­ pE

generate

Conversely, every locally convex structure

can be so generated since the gauge ("Minkowski functional") inf { /\

+ p( /\ x )

The closed semi­ball {x \ p(x )

balls (or eqUivalently the open semi­balls) pertaining to the a locally convex structure on

IR

Pv(x)

xE /\ V} of an absorbing, balanced, convex set

is a

V

semi­norm.

1.6. space.

Example.

Let

A Nachbin family

continuous functions on

X if

be a completely regular (Hausdorff) topological

on

X

is a collection of positive, upper semi­

X such that for

v

l,v 2

E":

11

and

:A> 0

(X) ) v(x), x EX. Denote by l(x),AV2 vector space of all continuous functions f on X such that

VE?f with max(Av

for all

v E lJ.

sup Iv(x)f(x)\

xEX

The family of semi­norms

(p) given by v VE 1f

defines a locally convex structure on

r

there exists

Z! (X)

vf

is bounded

P (f) v

e 1t" (X).

the

- 43 Particular cases:

a) if

is the collection of characteristic

functions of compact subsets of X,

t: 11 (X)

then

r::

is the space

(X)

of

all continuous functions. and the locally convex structure is that of "uniform convergence on compact sets"; b) i f ?r

..8

space

t: 7F (X)

is the set of all positive constants, then

is the

of' bounded continuous functions with the locally convex struc­

(X)

ture of uniform convergence on X;

'if

c) if

is the set of all positi ve , bounded, upper semi­continuous

e >

functions which vanish at infinity (1. e . , given set

such that

KC X

Iv(x)I

C

if

x¢ K), then

there exists a compact

0

e2J(X)

is J:3(X)

with

the strict topology of R.C. Buck and van Rooij; d) if X is locally compact and (i­compact and

e 7J(X)

of all positive. continuous functions, then

Lr

is the collection

is the space

of

J'{(X)

continuous functions with compact support, and its locally convex structure corresponds to the usual "inductive l:Lmit" topology (cf. 1.9).

t: '2J (X) and their vector­valued analogues consult

For properties of

the works of K.D. Bierstedt, W.H. Summers and the references quoted there.

1.7. space tures

Let 11) 1 and 1IJ 2 be two collections of subsets of the vector

E satisfying (1) ­ (4) of 1.1 and defining the locally convex struc-

oe 1

and

'7::2 ,

coarser than L 1)

ee­2 )

1.8.

respectively.

11)1 ::) 1f}2'

if

than L

is

(or

"'C'2

is

IE: (E, 'r'l)

is a morphism. Let

be a family of locally convex spaces; suppose that the

locally convex structure f...

"t'l

2 1. e., if the identity map

.or:

of E...

V... '

is given by

be a linear map from a fixed vector space

E

and for each

into E...

of all sets (or of the balanced, convex ones) of the form generates the coarsest locally convex structure on

The collection f;l(V... ),

subspace of

E

and

convex structure on

j:

a) Let E L c..;. E

be a locally convex space;

the canonical injection.

L with respect to

j

V... E hO '

E for which all the

are :morphisms, called the initial structure with respect to the Particular cases:

let

t:

f...

f.... L a linear

The initial locally

is the induced structure.

be a family of locally convex spaces, E = ITE ... ... the product th the'" projection. The initial of the vector spaces E... and b) Let

E.

locally convex structure on structure.

E with respect to the

pr...

is the product

- 44 1.9.

Let

be as in 1.8 and for each t.. let

(Et.). (ce:), (10.. )

a linear :map from

E..

into a vector space

ing, balanced, convex subsets

V of E

E.

f:l(V) E 19..

E

subspace of E

'f

and

ElL.

vector space

for all

for which all the

are 'morphisms, called the final structure with respect to the a) Let E

be

The collection of all absorb-

such that

generates the finest locally convex structure on Particular cases:

f..

be a locally convex space;

the canonical surjection from

E

I-

fl-

f .. L a linear

onto the quotient

l'

The final locally convex structure with respect to

is the quotient structure. b) Let E.

be a family of locally convex spaces,

ternal) direct sum of the vector spaces final locally convex structure on

EI-

and

j..

E

=

UE .. I­

the (ex-

the t..-th injection.

E with respect to the

jl-

The

is the locally

convex direct sum structure. 1.10.

Let E

the collection 1f)

of subsets.

'>. > 0

there exists subsets of

be a locally convex space whose structure is given by

E

(L) if

A set

ACE

A C A V.

such that

is bounded i f for every VE

The collection $

and AE

B C A, cover

d(;

B E Jr ,

then E,

(iii) any finite union of sets belonging to balanced, convex set belonging to

If

Ae

»

/\ > 0,

and

is contained in a

, /. A c:,r;

then

E is a vector space and

.

a collection of subsets of

satisfies (t ) - (Lv ) , we say that equipped

of all bounded

satisfies the following conditions: AE

(ii) the sets

(Lv ) i f

is a convex bornology on

E,

which

E

and

a convex bornology is called a convex bornological space.

refer to [29J

1IJ

E

He

and to the lectures of Hogbe-Nlend in the present volume for

the theory of these spaces and its application to locally convex spaces [29, Chap. VII and IX]. 2.

Duality 2.1.

A continuous linear form

continuous linear map from E lK

Izi

f

on the locally convex space

into lK,

1. e.,

a morphism

f: E

E is a It o,

for

0

y EM.

g(x)

I-

O.

p(x)

for

x E E and

g(y)

=

fey)

= E. Y = M.

and satisfies

g(x)

From g(A x ) follows that

p( A x ) g(/\ x)

g(rx)

and 0

x E E.

for

p(x)

integers. we have

g

y EM.

= g

(mx ) = mg(

E

x)

for r.

for every positive rational Le .• g(l\x)

pC-Ax).

Finally if

which is additive on

ng ( ; x)

Since

A> O. /\-+ 0

as

f

of

rg(x)

=

g(-i\x)

for all positive real

-pC >'(-x)).

and consequently

< O.

g(;\x)

=

it

l\g(x)

g( I\x) = g(-i\(-x))

then

;'g(x).

=

2.8.

JR. such that

g: E

for

f(y);;;; p(y)

U = E and condition (a) of 2.5 is satis-

Clearly

fied. so that there exists an extension

-Ag(-x)

= O. and such that

p(O)

Equip E with the strongest order and apply Aumann's theorem

(2.6) with X

>0

M a SUbspace of E.

JR. be linear and

f: M

Let

Then there exists a linear form

Proof.

m.n

E.

be a finite. subadditive function on

p( A x) for

Theorem.

A generalization of a well-known consequence of the Hahn-Banach

theorem is due to Konig. (2.8.1)

Theorem [40J.

subset of E and additive and

p

Let

a finite sublinear function on E.

p(ftx) = Ap(X)

is

z E A such that

on

E with e(x)

E be a real vector space,

for

p(z - i(x+y)) p(x)

for

O. O.

A a non-empty

i.e..

Assume that for

p

is sub-

x.yEA

there

Then there exists a linear form

xE E and such that

t

inf p (x ) = inf ((x).

»:.A

XEA

Fuchssteiner [26J shows that a generalization of (2.8.1) to semigroups follows from 2.4. (2.8.2) space

E and

there is

Corollary p

[40J.

Let

B be a non-empty subset of a real vector

a finite sublinear function on E.

z E B such that

pf z - (x+y)) ;a O.

If

Assume that for

p(x )

0

for all

x.yE B x E B.

- 49 -

t

then there exists a linear form

t (x )

such that

for all

0

Indeed. the set

p(x)

for

x E E and

x E B.

i

A '"

condition of (2.8.1).

on E with I(x) x E B.

n

1

integer} satisfies the

From these two results Kbnd.g [40J obtains simple

proofs in a number of situations. in same of which the minimax theorem was used earlier. as for instance the separation of convex sets. the existence of a Jensen measure. the fact that the Silov boundary is the closure of the Choquet boundary. the Hoffman-Wermer lemma. the Glicksberg and the KonigSeever generalizations of the F. and M. Riesz theorem and their equivalenoe due to Rainwater. and results of GrUnbaum J Kirszbraun and Minty on quadratio forms.

2.9.

Another oonsequenoe of (2.8.2) is the following maximal theorem: Theorem [41].

(2.9.1)

Let V be a real vector space and

(v)

empty oolleotion of sublinear functions on V such that

if (v)
0 ; we get (1) by renorming H+ • A vector f E H generates an antilinear functional 1 on H+ as o follows leu)

=

If(u)

= (f,u)o = (f,iu)o '

this functional is continuous since by (1) Ilf(u)!

=

I(f,iu)ol < [[flf 0 '1liull 0 Hk and satisfying the equation (Au,v)l (u,A*v)k' u E Hk, v E HI' For u E H_ l, u E Hk we have

spaces;

[u,Au]o

(9)

(rlu,Au)l A+ =

(A*Ilu,u)k

=

I k A *11 1

and we immediately see that II A+ 1/

=

[Ik1A* IIA*II

lU'u]o

= IIAII.

=

[A+u,uJ o' where

.. 166 -

:r

We know that maps H_ isometrically into H+; we now show that this operator may be factored into two operators, the first of which maps H

-

isometrically into H , and the second H 0

0

'I

into H+,

The operator I acts continuously from Ho to H+, Since H+ this operator may be as acting in Ho'

Ho

I)

J

We introduce the notation "I operator

I

=

iI for the latter operator, The

is obviously continuous, nonnegative

{(2)

(ilf,f)o

= (If,If)+ :::: 0 ) , and invertible on R(I) {If = 0

I1IJ 0 = (If,u)+ (f,iu) ) iu dense in H f = 0 ; i is invertible) , We will show ",0 -1 0 A. ... that R(I) = .:D(I ) is dense in Ho: i f h.LR(I), then 0 = (h,If)o = = (Ih,If)+ = (ilh,f) for any f E H , therefore ilh = 0, and hence 1 h = 0, It is clear is is selfadjoint!) and

=

1-

(I

positive in Ho' Theorem 1,

Consider the operator I) =

in the space Ho' It is a positive selfadjoint operator for which I

I

»(I)) = H+, R(I)) = Ho' This operator acts isometrically from H+ to Ho: (u,v)+

(I)u,I)v)o'

=

u,v E H+ •

Consider I) as an operator acting from Ho to Hand form the closure by continuity; denote this operator by ]) , ]) acts isometrically from all of Ho to all

H : moreover (10)

(f,g)o

()f,l>g)_,

=

])

0

f,g

E

Ho ' and

I)

The relation (11)

(f,I)u)o

=

' f

holds, from which it appears adjoints of each other ( ]) = I) )'

Ho' u I) and E

»

E

H+ J are

- 167 -

I- 1 •

Equation (10) gives a factorization of that

I

(12) Here

From this it follows

= D- 1 ])-1 or, if we introduce the operator J = D- 1 T:=Jo;;

'7

denotes ,])-1; thus (12) gives a factorization of

I

into

isometric operators ., and J. If we replace f by } a and u by Jf in (11), we obtain

(7 a , f) o

=

(a,Jf)o' a

E

H_, f

Ho ' Le • ., = J+.

E

We list the basic properties of the isometric operators:

(r a ,

(Jf,Jg)

(l a, (13) ,

(a,

( Du , Dv ) 0 = }

= J+,

.J> =

(u , v ) + ' D+ :

The inclusion i: H±

=

(,) f, J) g) _

(') a, f)

0

(f,g),

=

=

(1 (f, g ) 0

;

= (a, Jf), (J) f , u ) 0 = (r , Du)

Ho is H.S.,iff

J is

H.S.

(J

=

0

iJ, i =

•

jJ- 1=

3D),' equivalently; iff i': Ho H is H.S. We now show how to construct a rigging given an operator T.(H.S. = Hilbert-Schmidt operator.) Let T be a closed operator in a Hilbert space Ho' having a dense domain (T) and such that

=

(14 )

IlTullo::: [v

ll

uED(T).

Obviously D(T) is a pre-Hilbert space with respect to the scalar product ( 15)

(u,v)+ = (Tu,TV)o )

and we can take the completion of this space as a positive space H+ and then construct a corresponding negative space H • Consider the operator D with respect to the sequence H+

H C> H ; o equations (u,v)+ = (Du,Dv)o and (15) show that D and T are metri-

cally equal (if T is, in addition, positive, then obviously D = T). 1 On R(T), T- exists and is continuous; it is metrically equal to

. -1 " = J. Thus II.D

From this and the assertion above

it follows that the inclusand only if T- 1 is H.S. Notice that in

ion H+ ---> Ho is place of (14) it is possible to use the estimate IITull U E

D(T)

> eIIull

0-0

with C > 0, or what is equivalent, that the equation

,

- 168 -

T*x = f is solvable for any . f E H0 and x depends continuously on f. Example. We consider for simplicity G = R Let s,t E R and n n. W(x,t) = (-1) where (a,b); we put

n

.

n

slgn(xl' .... x) TT n j=1

is the characterizatic Du

u

1 x 1 · ··ax n

(t J.) , function of the intervall

, D+ = (_1)nD.

It is easy to verify that for f E Lioc (R ) and u(x) E we have n Dx fW(XoJ)f(J)df = f(x), = u(I)) (16)

n

n

n

c,,(x,J)(Du) (!)d!

u( x ) •

Let us define T: (Tu)(x) = q(x)(D+uXx),

u

E

'>

D(T) =

where q(x) = (1 + Ix11)1+i ... (1 + Ixnl)1+{ , O. Using the second of the formulas (16) we find that the left inverse -1 . T is anmtegral operator (Tand

1f)(x)

f f




are bounded operators on H. The function (6) is called a nonnegative operator-valued measure, if it takes the value zero on the empty set, if the operators

e (A)

are nonnegative and if they

- 169 -

satisfy the requirement of weak countable additivity: for disjoint the equation ( 17)

holds in the sense of weak convergence. Suppose that the measure for bounded

g

(A)

=

Ii:

tr

(0

(f)))
0, il existe une pcu Cf = (Cfi) telle que

IhUH

pour toute

(3.3.2) COROLLAIRE.­ La topologie de Moo(T) et la ¢­topologie coincident sur les bornes communs et sur Ie cone positif. En particulier elles \ ont les memes parties compactes. Or la

¢­topologie est une topologie initiale d'espaces

sorte que les proprietes speciales des espaces

1

de

(I) se transportent

aux espaces Moo(T). Ainsi (3.3.3) THEOREME.­ Toute suite de Cauchy faible dans Moo(T) est conver!gente dans Moo(T). En particulier Moo(T) est faiblement semi­complet. (3.3.4) THEOREME.­ Pour toute partie s on t equivalentes : / a) A est relativement compacte ;

les assertions suivantes

- 198 -

b) A est relativement faiblement compacte ; c) A est b or-nee et, pour toute pcu f = ('f'i)' les familIes decrit A. = sont equisommables lorsque En suivant Haydon [H 1], et par un assez joli lemme technique, on en tire : (3.3.5) COROLLAIRE.- L'enveloppe solide IAI d'une partie relativement Icompacte A de Moo(T) est encore relativement compacte. 3.4 Questions d'ordre. La situation de l'espace MOO(T) a l'interieur de l'espace M(ST) des mesures de Radon sur ST est remarquable vis-a-vis des proprietes d'ordre. On sait deja que M(BT) est un espace de Riesz completement reticule. Or pour toute et toute pcu f = on a = alors que l'egalite a lieu pour toute (T) puisque Moo(T). Mais [R31 : 00

(3.4.1) THEOREME.- Pour qu'une mesure i I faut et il suffit que l'on ait pcu f = sur T.

I

soit element de Moo(T), = pour toute

Fixons HEClf, Het., et E >0 ; on sait deja qu t i L existe une p cu

=

et une famille de points de T telles que Qf-L:f(ti)CfilkE pour toute r s n. Donc en fixant telle que et en choisissant la partie finie J de I pour que L I u I E , on a ifJ l pour toute f E. H L

ilaJ

f (t . ) u (CD. ) l

'l

I(

E

Or L

ifJ donc

cp.) =

L

L

i J

l

to.)

i£J \

l

l

l

= pour toute fEH,

ce qui exprime encore que :

-

Et i II H' 2 E

et montre que est limite uniforme sur H de mesures discretes est donc continue sur H et Moo(T).

elle

- 199 -

D'ou l'on tire:

(3.4.2) COROLLA1RE.- Moo(T) est une bande dans l'espace completement re-

I

M( 8T) .

11 est facile de voir que Moo(T) est solide dans M(8T). Pour le reste, il suffit de voir que toute mesure positive u e. M(8T), borne s upe r-Leu r-e d'une famille filtrante croissante de mesures positives Moo(T), est element de Moo(T). Or pour chaque pcu = on a : = Sup u (1) = Sup Sup

L: (cp.) = Sup J iEJ Il J ou J decrit l'ensemble des parties finies de I.

L:

l

=

l

On peut terminer en caracterisant la bande etrangere a Moo(T) dans M(8T). Pour cela associons a chaque mesure positive M(8T) et a chaque pcu

= la mesure = L: On a et, de plus, la famille est filtrante decroissante, car si 1)J = (1)Jk) est une autre pcu, la famille = (tfi1)Jk) est aussi une pcu et u'f'l',I, = . Par ailleurs 'f' si ve. M (T) alors v'f = v, de sorte que lIon a pour toute ve.M (T) telle que . 11 suit de la que la mesure A = Inf majore toute 'f v EM oo( T ) telle que En f'a i. t : 00

00

(3.4.3) THEOREME.- La mesure A Ila composante de

= 1nf

est element de Moo(T) et c'est

'f

00

sur la bande M (T).

=

Posons pour simplifier

L:

iEJ

l

pour toute partie finie J de I.

11 suffit de prouver que l'expression L

L

= Inf Inf

J

= Inf

1)J

mais

1)J

=

=

=

est nulle. Or

J

=

J A(l), d'ou

1)J

J = O.

0.4.4) COROLLA1RE.- Une mesure

est etrangere a la bande Mco(T) s i et seulement s I il existe, pour tout 00, une pcu 'f = ('fi) sur T

l

telle que

On remarquera qu'on generalise ici au cas de Moo(T) 4n critere classique de Tamano [Tl selon lequel un c ar-ac t er-e u e. 8T est e x t e r-Leur' a 8T (ou etranger a Moo(T)) si et seulement s'il existe une pcu telle que

=

0 pour tout i.

- 200 BIBLIOGRAPHIE (B 1)

N. BOURBAKI, Integration, chap. IX, 1969, Paris.

(B 2)

H. BUCHWALTER, Topologies et compactologies, Publ. Dep. Math. Lyon, 6-2, 1969, p. 1-74.

(B

3)

(B 4)

H. BUCHWALTER, Parties bornees d'un espace topologique completement regulier, Seminaire Choquet, ge annee, 1969-70, n014, 15 p. H. BUCHWALTER, Sur le theoreme de Nachbin-Shirota, J. Math. pures et appl.,

a

paraitre en 1972.

Voir aussi : Comptes rendus, t. 273, serie A, 1971, p. 145-147. Comptes rendus, t. 273, serie A, 1971, p. 228-231. (B

5)

H. BUCHWALTER, Espaces ultrabornologiques et b-reflexivite, Publ. Dep. Math. Lyon, 8-1, 1971, p. 91-106.

(BI 1) J. BERRUYER-B. IVOL, Une topologie sur l'espace des mesures de Riesz, Comptes rendus, 274, serie A, 1972, p. 1927. (BI 2) J. BERRUYER-B. IVOL, L'espace M(T), Comptes rendus, 275, serie A, 1972, p. 33. (BI

3)

J. BERRUYER-B. IVOL, Espaces de mesures et compactologies, Prepublications St-Etienne, 1972. A paraitre aux Publ. Dep. Math. Lyon,

(BN)

H. BUCHWALTER-K. NOUREDDINE, Topologies localement convexes sur les espaces de fonctions continues, Comptes rendus, 274, serie A, 1972, p. 1931.

(BP

1)

H. BUCHWALTER-R. PUPIER, Caracterisation topologique de la completion universelle d'un espace topologique completement regulier, Comptes rendus, 268, serie A, 1969, p. 1534.

(BP

2)

(BS)

H. BUCHWALTER-R. PUPIER, Completion d'un espace uniforme et formes lineaires, Comptes rendus, 273, serie A, 1971, p. 96. H. BUCHWALTER-J. SCHMETS, Sur quelques proprietes de l'espace Cs(T), J. Math. pures et appl.,

a

paraitre en 1973.

Voir aussi : Comptes rendus, 274, serie A, 1972, p. 1300. (C)

G. CHOQUET, Cardinaux 2-mesurables et cones faiblement complets, Ann. Inst. Fourier, Grenoble, 17-1, 1967, p. 383-393.

(D)

R. M. DUDLEY, Convergence of Baire measures, Studia Math., 27, 1966, p. 251-268.

(DJ)

J. DAZORD-M. JOURLIN, Sur quelques classes d'espaces localement convexes, Publ. Dep. Math. Lyon, 8-2, 1971, p. 39-69.

- 201 -

(GJ)

L. GILLMAN-M. JERISON, Rings of continuous functions, Princeton, 1960.

(H 1)

R. HAYDON, "Without title", Cormnunication personnelle, 1972,

(H 2)

E. HEWITT, Rings of real-valued continuous functions I, Trans. Amer.

Cambridge.

Math. Soc., 64, 1948, p. 45-99. (K)

Y. KOMURA, On linear topological spaces, Kumamoto J. of Sc., serie A, 5, 1962, p. 148-157.

(LS)

C. LEGER-P. SOURY, Le convexe topologique des probabilites sur un espace topologique, J. Math. pures et appl., 50, 1971, p. 363425.

(N 1)

L. NACHBIN, Topological vector spaces of continuous functions, Proc.

(N 2)

K. NOUREDDINE, L'espace infratonnele associe

(R

(R

(R

1)

2)

3)

(S)

Nat. Acad. Sc. USA, 40, 1954, p. 471-474.

a

un espace localement

convexe, Comptes rendus, 274, serie A, 1972, p. 1821. A. ROBERT, Quelques questions d'espaces vectoriels topologiques, Cormnent. Math. Helvet., 42, 1967, p. 314-342. M. ROME, Le dual de l'espace compactologique

Comptes rendus,

274, serie A, 1972, p. 1631. M. ROME, Ordre et compacite dans l'espace Moo(T), Comptes rendus, 274, serie A, 1972, p. 1817. T. SHIROTA, On locally conVex vector spaces of continuous functions, Proc. Japan Acad., 30, 1954, p. 294-298.

(SW J. SCHMETS-M. De WILDE, Caracterisation des espaces C(X) ultrabor1) nOlogiques, Bull. Soc. Roy. Sc. Liege, 40e annee, 3-4, 1971, p. 119-121. (SW J. SCHMETS-M. De WILDE, Locally convex topologies strictly finer 2) than a given topology and preserving barrelledness or similar properties, Bull. Soc. Roy. Sc. Liege, 40e annee, 3-4, 1971, p . 119-121, (T)

H. TAMANO, Some properties of the Stone-Cech compactification, J. Math. Soc. Japan, 12, 1960, p. 104-117.

(V)

V. S. VARADARAJAN, Measures on topological spaces, Amer. Math. Soc. Translations (2), 48, 1965, p , 161-228.

- 202 -

(W1)

S. WARNER, The topology of compact convergence on continuous functions spaces, Duke Math. J., 25, 1958, p. 265­282.

(W 2)

R. F. WHEELER, The strict topology, separable measures and paracompactness, a paraitre au Pacific J. Math. en 1973.

(WH)

M. De WILDE­C. HOUET, On increasing sequences of absolutely convex sets in locally convex spaces, Math. Ann., 192, 1971, p. 257­261.

Convergence Structures on

E. Binz

Given a completely regular topological space X, it is well known that

the IR-algebra of all real-

valued continuous functions of X does not determine X. This means, two completely regular topological spaces Y and Z for which

is isomorphic to

(as

unitary IR-algebras) need not to be homeomorphic. In fact, the canonical map pactification

vX

i

x

from X into its realcom-

sending each point

p E.X

into its

point evaluation - called charactere by Buchwalter induces an isomorphism i

sending each

[121

*x

: ce.(vX) __ C0X),

ge cec"X)

into

g

Q

i

X

. One might consult

for examples of completely regular spaces being not

realcompact. However, equipping

and

both with the

topology of compact convergence yielding the topological algebras

and

homeomorphic iff

the spaces and

Y

and

Z

are

are bicontinuously

isomorphic. But in investigating the relationship between X and eco(X)

one has to deal with two unfortunate facts, namely: is in general not complete and in addition

the evaluation map

- ?04 -

co

(X)(

x -*

lR

W

\V

(f, p)

f(p)

is not continuous (with respect to the product structure). Moreover, there is in general no vector space topology T on

for which IJJ :

X - + IR

is continuous. We will see this later on.

[4J )

But one finds always (complete [7], structures on

convergence

making the evaluation map continuous

(simultaneously in both variables)

[9] .

Let me briefly repeat the notion of a convergence structure. For a given set W, we denote by filters on W. A map of

A

F(W)

the set of all

from W into j'(F(W)), the power-set

F(W), is a convergence structure, if for each point

PEW

the following axioms are satisfied:

(i)

p, the filter generated by all supersets of

{pj)

belongs to A (p). (ii)

Any filter'±'

finer than a filter

4E.

A(p)

belongs

to !\(p). (iii)

The infimum of two filters to

/\(p)

belongs

A(p).

A set W endowed with a convergence structure

A

called a convergence space. The filters belonging to are called the filters converging to

p

E.

W.

is

1\

(p)

- 205 -

Given a convergence space W. The fact that a filter

'

on

' ((3X\K)

converging

for some K.

The convergence structure, called I, defined in this way allows W

:

to be continuous, which means that

is continuous.

X-- lR.

- 208 -

The convergence structure just defined has the following characteristic universal property: For any topological ffi-vector space E a linear map I into

is continuous with respect to I iff it is

continuous with respect to If

I: E

Ac•

is continuous, then

for some compact

I(E) c

\X.

Kc

Using the universal property of

one quickly

deduces that given a vector space topology T on making id : set

W;

iPT (X)

x

X ---+

--. K c:.

X\X

IR

continuous, implying that,

is continuous, there exists a compact such that Xc

Hence if X is realcompact, such a topology T on requires X to be locally compact. On

therefore1we do

not find any vector space topology T for which W ;

- - IR

is continuous. The next theorem

[5J

and the following

comments indicate

parts of the aim, I intend to reach:

Theorem Let X be a completely regular topological space. The convergence algebras

ec (X)

and

It?r(X)

are identical iff

the following two conditions hold: a)

The intersection of countably many neighbourhoods of

X in

is a neighbourhood of X again.

- 209 -

b)

The points in X having no compact neighbourhood in vX form a compact set.

In fact,

(h) holds, iff tel (X) =

(X).

Condition (a) characterizes the identity of

ec (X)

and

The index u indicates the convergence structure of local uniform convergence. The last two results can be found in [13] ,where many more results of this type are presented.

Ref e r e n c e s

[5]

[7]

E. Binz, H.H. Keller

Funktionenraume in der Kategorie der Limesraume. Ann.Acad.Sci.Fenn. AI. 383, 1-21 (1966).

E. Binz, K. Kutzler

Uber metrische Raume und Cc(X). Ann.Scuola Norm. Sup. , Pisa. Vol. XXVI, Fasc.I, 197-223 (1972).

E. Binz, W. Feldman

A Functional Analytic Description of Normal Spaces. Can.J.Math., Vol.XXIV, No.1, 45-49 (1972)

E. Binz, W. Feldman

On a Marinescu Structure on C(X). Comm.Math.Helv.,Vol.46, Fasc.4, 436-450, (1971).

E. Binz, P.Butzmann W. Feldman, K.Kutzler and M. Schroder

On w -admissible Vector Space Topologies on C(X). Math.Ann. 196, 39-47 (1972).

E. Binz

Zu den Beziehungen zwischen c-einbettbaren Limesraumen und ihren limitierten Funktionenalgebren. Math.Ann. 181, 45-52 (1969).

E. Binz

Notes on a Characterization of Function Algebras. Math.Ann. 186, 314-326 (1970).

- 210 -

[8]

E. Binz

Kompakte Limesraume und limitierte Funktionenalgebren. Comm.Math.Helv., Vol.43. 195-203 (1968).

[ 9]

E. Binz

Recent Results in the Functional Analytic Investigations of Convergence Spaces. To appear in Proceedings of the Third Prague Topological Symposium.

[10]

W. Feldman

Topological Spaces and their Associated Convergence Function Algebras. Ph.D.Thesis. Queen's Univ .• Kingston.Can.

[11]

H.R. Fischer

Limesraume. Math.Ann. 137, 269-303 (1959).

[ 12)

L.E. Gilman. M. Jerison

r13]

Rings of Continuous Functions. van Nostrand. Princeton 1960.

K. Kutzler

tiber Zusammenhange. die zwischen elnlgen Limitierungen auf C(X) und dem Satz von Dini bestehen. Habilitationsschrift, Univ. Mannheim (1972).

[14}

B. Muller

tiber den c-Dual eines Limesvektorraumes. Dissertation, Univ. Mannheim (1972).

- 211 -

VARIOUS TYFES OF BARRELLEDNESS AND INCREASING SEQUENCES OF BALANCED AND CONVEX SETS IN LOCALLY CONVEX SPACES M.DE WILDE Various tjpes of barrelledness

1.

We use the notations of Horvath's notes in this volume. A locallj convex Hausdorff space

E is quasi-barrelled (a-barrel-

led) if everj a(E'. E)-bounded set of of equicontinuous subsets of

E'

which is a countable union

E' (a countable subset of

E') is equi-

continuous. The space

E is quasi-infra-barrelled (a-infra-barrelled) if

everj B(E'. E)-bounded subset of equicontinuous subsets of

E'

which is a countable union of

E' (a countable subset of

E') is equiconti-

nuous. The reason for considering these various tjpes of barrelledness finds its origin in the following proposition. due to Grothendieck [51. Froposition 1. for

If

E is metrizable. E'

is quasi-infra-barrelled

B(E'. E). An interesting problem was to find conditions impljing that

is even barrelled for in [5].

B(E'. E).

E'

It has been studied bj Grothendieck

Actuallj. Grothendieck has substituted to the duals of metri-

zable spaces. a new abstract class of spaces. the so-called "'!)J'-spaces". with the following properties: E is a

if

(a)

E is quasi-infra-barrelled.

(B)

E admits a fundamental sequence of bounded sets. It is worth noticing that l.Schwartz has provided a completely dif-

ferent approach to the barrelledness of strong duals of usual spaces. It is developped in Hogbe's notes in this volume. In [7]. Valdivia has given several new properties of barrelled

- 212 -

spaces.

It was clear that some of thse properties were inspired bj

Grothendieck's theorj of

although not including it.

The

gap between [51 and [7] was filled in [31 and. s ur pr i s t nql y , the gain of generalitj came along with a gain of simplicitj. matter of this lecture.

That will be the

The content is essentiallj the same as in [31.

presented in a somewhat more general setting. Absorbent and bornivorous sequences

2.

let

E be a locallj convex Hausdorff space.

An absorbent (bornivorous) sequence is an increasing sequence

em

of convex and balanced sets. such that each element (bounded set) in E is absorbed bJ one of the sets

em'

All the results included in these notes for absorbent sequences are also true for bornivorous sequences. when replacing the barrelledness assumptions bj the corresponding infra-barrelledness assumptions. Theorem 2.1.

let

l

an absorbent sequence in (H)

for each sequence

the set

mE

}

x'm E E'

U:=l em

l .. E.

Then

Thus e

(m

E

< x. x' > ..

1 +

xm I E eO m for each

x

(l+ )Uem.

< x , x'm > l.

m,

U:=l em'

Then. for each and

1 +

It is not a restriction to assu-

m E IN}

It has therefore an adherent point it is clear that

em

is the algebraic closure of

such that

The sequence is equicontinuous in me that

E and

such that

> O. assume that

Given

m, there exists

l

xm I (m E 1N) such that is equi conti nuous.

Then the closure of Froof.

be a vector subs pace of

is also equicontinuous in x'

for

and that

aCE'. E). x'

E

For that

eOm for each

E. x'. m.

x

(Uem)oo .. Ue m. BJ the obvious correspondance between the Cauchj filters on a set

and the points of its completion. theorem 2.1 can easilj be inter-

preted in a form similar to lemma 3.9 in Horvath's notes.

- 213 -

There are two important examples of absorbent sequences verifJing (H) •

(a)

If

spaces

L is the inductive limit of an increasing sequence of sub-

Lm• then If

(8)

em

= Lm (m

verifies

E

(H).

L is a-barrelled. each absorbent sequence in

L verifies

(H) •

Corollaries 2.2.

A strict inductive limit of complete spaces is complete

(Kl5the). Froof.

Take for

the inductive limit and for

L

E

its comple-

tion. 2.3.

r

If

is a-barrelled and has an absorbent sequence of com-

plete subsets. it is complete. 2.4.

If

Take

L

rand

=

E is a Baire space and if O.

em (m E

one of the

In particular. E is then barrelled.

It is worth noticing that

E is a Baire space when

zable. Without the strong assumption that get an interesting property.

= r.

E is a-barrelled. then.

for each absorbent sequence of closed sets em's is a neighbourhood of

E

E is metri-

.

E is a Baire space. we still

Before stating it. let us improve theo-

rem 2.1 by a trivial remark. that

If

em (m

E

Ue m

and

E

= ue'm'

then

e'm (m E

are increasing sequences such

E

= U( em n

.

Now, in theorem 2. I, assume that sequence in

where

em

e'm

E

such that

IN) is an absorbent

E

m.

e'm ue m = alg. cl. (Uem ) denotes the closure of

topologJ induced bJ Froof.

em E

e'm (m e'm for each

It

E.

is obvious. since

em

e' - m em

in

.

Then

e'm equipped with the

em n e'm'

- 214 2.5.

Let

E be a-barrelled and

be an absorbent sequence in

E.

A balanced, convex and absorbent set

a is algebraically closed. E is barrelled, and if a ('\ e'm is closed for

e ('\ e'm is closed in each

if

In particular, if each

m, a

e'm and

is a neighbourhood of

Froof.

a is a barrel if and only

O.

In the improved version of theorem 2.1. take

a ('\

em

Compare with proposition 3.10 of Horvath's notes. 2.6.

[5] If

E is metrizab1e. the following are equivalent

ex.

Ea (E ' • E)

is bornologica1.

B.

Ea(E' • E) ES(E'. E)

is u1trabornological.

y.

Since

is barrelled.

ES(E'.E) .. ex. If y

is

complete. ex

6 .. y.

a is bornivorous and if Urn nei ghbourhoods of 0 in E. e contains a' r( u Now

00

b1e

Am's.

m=1

We can write

a

is a basis of for suita-

I

N

Since

is a bornivorous sequence of closed subsets. m=1 by theorem 2.1 and proposition 1. the closure of a' is its algebraic closure.

3.

r( u

Thus

a

Theorem 3.1.

sed sets in

e'

Let

2

a'. the last member being a barrel in

em (m

be an absorbent sequence of clo-

E

E.

Each strongly bounded set in - If (m

E

em (m

E

verifies

E is absorbed bJ some

(H). (see theorem 2.1). then

IN) is bornivorous. ( [3].

th. 1. ) .

There is a kind of "dual" statement.

em' em

- 215 let 1.1

Theorem 3.2.

be a subset of :feE)

be a sequence of sets such that. if {x

m

: mE

for each

e is balanced and absorbs the elements of

If some

:B.

} E

xm E em

and let

em (m

IN)

E

m. it absorbs

em Corollaries. 3.3.

spaces

If

E is the inductive limit of an increasing sequence of

n

Em' ever) bounded set

(closure in

E).

If the

Em

is contained in some

E

are normed. jB is contained in the clo-

sure of a bounded set of some 3.4.

in

Em'

Under the same assumption on

and absorbs the equicontinuous sets of 3.5.

If the

E'm.s (Em.E ' m)

Em

E'

is balanced EOm for some

m.

E'

is bornological and is the .

3.4 and 3.5 are slight generalization of

r 51,

lemma 5 and

can be found in [21, prop.lO-lI, p.139-l40.

4.

also provides some interesting properties. Theorem 4.1.

E is quasi-barrelled and i f

If

absorbent sequence in U

Cf.

in

0

em (m E tl)

E = Ue m, a balanced and convex in E i f and onl) i f U n em is a

em

0

for each

m.

r 3] , theorem 3.

Theorem 4.2. subspace of

If

E is a-barrelled and

E. then each barrel in

l

a separable vector

E induces a neighbourhood of

l.

Cf.

r 3] • theorem 4.

Corollaries. 4.3.

is an

E such that

is a neighbourhood of

neighbourhood of

in

e

E'. then

E'S(E',E)

projective limit of the spaces

set

C

are moreover quasi-barrelled and the

bornological. then -

theorem 10.

e

E. if

If

E is a-barrelled and separable, it is barrelled.

0

- 216 4.4.

Let

its barrels. in

t

be the loca11j convex topo1ogj defined on

Then, if

E

E £l

a-barrelled, a sequence is convergent

Et if and on1j if it is convergent in E. Froof. Take for L the linear hull of such a sequence and its

limit and app1j 4.2. If

4.5. (m

E is quasi-barrelled and has an absorbent sequence

em

of metrizab1e subsets, it is barrelled.

E

Froof.

Bj 4.4, a barrel

each

e induces a neighbourhood of

Bj 4.1, it is thus a neighbourhood of 4.6.

[5]

bourhoods of

If

a

in

a

in

0

in

E.

E is metrizable and i f the po1ars of the neighE are metrizable for

R(E', E), then

ES(F',E)

is barrelled. Froof. po1ars. 5.

There is a bornivorous sequence of

E', made or such

The conclusion follows then from 4.5 and proposition 1.

Just like the concepts of barre11edness and infra-barre1ledness

extend to topological vector spaces, the definitions of § 1 can also be adapted to topological vector spaces, as well as absorbent and bornivorous sequences.

Most of the preceding results can be extended

although their formulation becomes rather sophisticated.

(see [4]).

REFERENCES 1.

AMEMI1A, I., KOMURA, 1.

Uber

Math.Ann., 177,273-277 (1968). 2.

DE WILDE, M. : Reseaux dans 1es espaces lineaires Mem.Soc.Roja1e Sc.Liege,

3.

a semi-normes.

2 (1909).

DE WILDE, M., HOUET, C. : On increasing sequences of abso1utelj convex sets in t ocel l y convex spaces. (1971).

Math.Ann., 192,257-261

- 217 4.

DE WILDE, M., GERARD-HOUET, C. :'Sur les pr opr t e t es de tonnelaqe des espaces vectoriels topologiques.

Bull .Soc.Ro}ale Sc.Liege,

11-12, 555-560 (1971). 5.

GROTHENDIECK, A. : Sur les espaces

(1)

et

Summa Brasil.

57-123 (1954). 6.

KOTHE,

u.

:

Topologische lineare Raume. I. Berlin-Heidelberg-

Springer, 1966. 7.

VALDIVIA, M. : Absolutel} convex sets in barrelled spaces. Ann.Inst.Fourier, 21 (1971).

LATTICE BOUNDING r1APPINGS

D.J.H. Garling

E

If

is a Banach space and

o

< p < (0) , y

f

E

E

LP W, u ) F

and

L(E,L P(S1,j.J) ) (for some

E

is said to be p-lattice bounded if there exists

/y (e)

such that

I

< f

are Banach spaces,

p-lattice bounding if y

y

yu

u

whenever L(E,F)

E

II

ell < 1-

is said to be

is p-lattice bounded whenever

(for arbitrary topological space

E

If

n

and regular

probability measure It turns out that p-lattice bounding mappings are closely related to p-absolutely summing and p-Radonifying mappings.

The

first two theorems are direct ones. THEOREM 1.

If

Y

is p-lattice bounded, then

£

y

is p-absolutely summing. THEOREM 2.

Suppose that either

p :

0

2

then

and u

is

q




p,

the diagonal mapping

summing if and only i f

ex E R,p

(when

p.:. r.:. q)

(when

q.:. r).)

Tong [8]

and

ex E R,q

0 < r

d

R,q , except

into

(For example i f is r-absolutely

ex

< p) , ex

R,r (when

E

Using ideas introduced by

(who characterised I-nuclear diagonal mappings), we can

then characterise all the r-integral and r-nuclear diagonal mappings, except for the case when

1 < r < min(p,q) .:. max(p,q) < 2.

It

turns out that (with the exceptions mentioned above) a diagonal mapping from

R,P'

into

R,q

r-absolutely summing, when into

of

is r-integral if (and only if) it is r

Is every

u

CONCLUDING REHARKS. of KwapieJ'l y

1.

In particular, the

r > 1.

is r-integral, for

summing diagonal mapping from r > I?

>

E

L(R,1,R,2)

p'

Is every r-absolutely r-integral, when

into

p-decomposable in the sense that there exist mapping of

each

e

into

E'

such that

is in

>

I?

Theorems 1-5 are closely related to results

is p-lattice bounding, then


0

0

if and only if there exists

be a linei'

F)

->

be an unbounded F)

->

a

be a non null

being a Hausdorff topological vector space. Then ->

r)

rV

C b

[resp G(f : b

a

->

a

F) C

and these inclusions are c ont.i nuoua ,

PROPOSITION

4.3. (cf [5J, proposition 2,b). Let

tp

be an unbounded

concave Orlicz function. Then bicontinuously where

i

tp

is the canonical injection

Ltp

->

L

O ,LO

is the space of all measu-

rable functions with convergence in measure.

Now, let us prove theorems 3.1 tinuous linear mapping

->

Ltp (resp b

a

a

continuously. So,

Itp C

rJ

f

be a non null con-

....

N

'"

Let

and 3.2

(resp (IJ

(resp some zero-neighborhood of

Itp

is bounded in

- 226 -

('I

It);) u

and, consequently,

(resp lim sup

0)

when

0 . This implies conch tions (i), and therefore (ii), cf theorem 3.1 (resp 3.2)

(of [5J) . BIBLIOGRAPHY

[1J

A. FROLICHER and W. BUCHER, Calculus in vector spaces without norn, Lecture Notes in Mathematics 30, Springer, 1966

[2J

W. ORLICZ, On spaces of functions, Proc. Intern. Symp. on Linear Spaces (Jerusalem, 1960), 357-365.

[3J

D. PALLASCHKE, The compact cnd omorpii s.cs of t':",11etric linear spaces

[4J

S. ROLEWICZ, Some

preprint. on the spaces

N(L)

and

N(1), Studia Math.18

(1959), 1-9. [5-]

P. TURPIN, Operateurs Li.nea i r c s en t r o e s j.ac e s d ' Crlicz non LocaLemen t convexes, Studia 46(2), to

[6 J

P. TURPIN, Sur t.n pro bl ome de S. SimlJns concernan t les bor-nes des espaces vectoriels topologiques, Bull. Soc. Math. Fr. (Memoires), to appear.

Lecture Notes in Mathematics Comprehensive leaflet on request

Vol. 146: A. B. Altman and S. Kleiman, Introduction to Grothendieck Duality Theory. II, 192 pages. 1970. OM 18,Vol. 147: D. E. Dobbs, Cech Cohomological Dimensions for Commutattve Rings. VI, 176 pages. 1970. OM 16,Vol. 148: R. Azencott, Espaces de Poisson des Groupes Localement Compacts. IX, 141 pages. 1970. OM 16,Vol. 149: R. G. Swan and E. G. Evans, K-Theory of Finite Groups and Orders. IV, 237 pages. 1970. OM 20,Vol. 150: Heyer, Dualitiit lokalkompakter Gruppen. XIII, 372 Seiten. 1970. OM 20,Vol. 151: M. Demazure et A. Grothendieck, Schemas en Groupes 1. (SGA 3). XV, 562 pages. 1970. DM 24,Vol. 152: M. Demazure et A. Grothendieck, Schemas en Groupes 11. (SGA 3). IX, 654 pages. 1970. DM 24,Vol. 153: M. Demazure et A. Grothendieck, Schemas en Groupes Ill. (SGA 3). VIII, 529 pages. 1970. DM 24,Vol. 154: A. Lascoux et M. Berger, Variates Kahleriennes Compactes. VII, 83 pages. 1970. DM 16,Vol. 155: Several Complex Vartables I, Maryland 1970. Edited by J. Horvath. IV, 214 pages. 1970. DM 18,-

Vol. 178: Th. Brocker und T. tom Dteck, Kobordismentheone. XVI, 191 Setten. 1970. DM 18,Vol. 179: Semtnatre Bourbaki - vol. 1968/69. Exposes 347-363. IV. 295 pages. 1971. OM 22,Vol. 180: Seminaire Bourbaki - vol. 1969/70. Exposes 364-381. IV, 310 pages. 1971. DM 22,Vol. 181: F. DeMeyer and E. Ingraham, Separable Algebras over Commutative Rings. V, 157 pages. 1971. DM 16.Vol. 182: L. D. Baumert. Cyclic Dtfference Sets. VI, 166 pages. 1971. DM 16,Vol. 183: Analyttc Theory of Differential Equations. Edited by P. F. Hsieh and A. W. J. Stoddart. VI, 225 pages. 1971. DM 20,Vol. 184: Symposium on Several Complex Vartables, Park City, Utah, 1970. Edtted by R. M. Brooks. V, 234 pages. 1971. DM 20,Vol. 185: Several Complex Vartables II, Maryland 1970. Edtted by J. Horvath. Ill, 287 pages. 1971. DM 24,Vol. 186: Recent Trends in Graph Theory. Edited by M. Capobianco/

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