Closed Graph Theorems and Webbed Spaces

MDeWilde University of Liege

Closed graph theorems and webbed spaces

Pitman LONDON · SAN FRANCISCO · MELBOURNE

PITMAN PUBLISHING LIMITED 39 Parker Street, London WC2B 5PB FEARON-PITMAN PUBLISHERS INC. 6 Davis Drive, Belmont, California 94002, USA Associated Companies Copp Clark Ltd, Toronto Pitman Publishing New Zealand Ltd, Wellington Pitman Publishing Pty Ltd, Melbourne

First published 1978

AMS Subject Classifications: (main) 46A02, 46A30, 46F02 (subsidiary) 46A05, 46F05

© M De Wilde 1978 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,or transmitted in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the publishers. The paperback edition of this book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form ofbinding or cover other than that in which it is published, without the prior consent of the publishers. Reproduced and printed by photolithography in Great Britain at Biddies ofGuildford ISBN 0 273 08403 8

Preface

The first closed graph theorem was proved by Banach for

~-spa­

ces, more than forty years ago. It became very quickly a central result in functional analysis and its applications. It has by now been generalized in many ways. The various generalizations essentially split in two main trends. On one side, the domain space is, for instance, assumed to be barrelled (or somethin~

connected to the Baire category property) and suitable

conditions on the range space are determined to insure the validity of the closed graph theorem. They turn out to be connected somehow to a completeness condition. A detailed account of these results can be found in [90]. Of course, for practical purposes, it is important to know whether the closed graph theorem is true for those locally convex spaces that occur in applications, such as the various spaces of distributions of L. Schwartz. The question was asked by Grothendieck in his thesis [45,p.19].

Observing that ultra-

bornological spaces would be a reasonably wide class of domain spaces, he conjectured the existence of a corresponding appropriate class of range spaces. The problem of finding an adequatly comprehensive class of range spaces was studied by Slowikowski [98,99], Raikov [87], L. Schwartz [94], Martineau [72,73,74] and the author [14,19,21]. The present notes are essentially devoted to the study of the author's webbed spaces and of the results of L. Schwartz and Martineau. The basic results on the domain and range spaces are developped in a self consistent way. A large part of the book is of graduate level. It also includes more specialized results, reaching the level of current research.

Chapter I begins with the closed graph theorem of Banach. Starting with the Baire category property, it develops the subject and its first easy extensions in a manner that traces the way towards further generalizations. Chapter II is concerned with the permanence properties of the domain and of the range spaces. The introduction of linear relations and the corresponding hereditary properties on the range space are due to Raikov. The reader who wants to restrict himself to the main results may omit this chapter, excepting prop.II.).2, which points out the interest of ultrabornological spaces as domain spaces. The theory of ultrabornological spaces is developped in Chapter III. Since it is entirely parallel to the study of bornological spaces, both cases are handled together. It is widely inspired by §28 of [62] and completed by results of Schwartz, Hogbe-Nlend and Grothendieck on ultrabornological dual spaces. Chapters IV and V present the theory of webbed spaces. We essentially follow [14], with various improvments due to the systematic use of fast convergent sequences. The basic results are in Chapter IV. Chapter V collects sophisticated permanence properties of spaces of linear maps, spaces of vector-valued functions and tensor products. Although the concept of webbed spaces has been extended to topological vector spaces, we restrict ourself to locally convex spaces. The reader will find in [90] an account of the theory for topological vector spaces. The fact that a linear map T is relatively open or weakly open, or that its range is closed, is closely related to similar properties of its adjoint T* and also to the possibility of lifting equicontinuous sets to equicontinuous sets by T*. In suitable cases, this lifting problem can be solved by means of webs. Chapter VI outlines these questions, which play an important role in applications as shown in [104,105].

The last chapter is devoted to the Borel graph theorem of L. Schwartz. After recalling the required properties of Souslin spaces, we present the proof of Martineau and another one due to Christensen. The results are then extended to topological groups and various permanence properties for spaces of linear maps are sketched. I want to conclude by thanking J. Schmets for his helpful remarks and careful reading of the manuscript. I am also indebted to Mrs Streel for typing the first draft and the camera ready manuscript. Marc De Wilde

Terminology and notations

The main part of these notes is concerned with Hausdorff locally convex topological vector spaces (l.c.s). Our notations are pretty standard. We call an U-space a metric complete l.c.s. A

L~-space

is a countable inductive limit of such spa-

ces. A countable inductive limit of l.c.s. E. (i E-lN) is strict ~

if the topology induced by Ei+ 1 in Ei coincides with the topology of E ~.• Let E be a l•c.s. If A C E, we denote by Co A, Co A and >A

n nE-lN

wn 1=¢ •

Indeed, assume that a non empty open subset w is meagre, hence contained in the union of ~ountably many closed subsets Fn(nE-lN) with void interior. Since F1 =¢,we have w j F 1 • There exists thus an open subset w1 .f=¢ of w such that w1 n F 1 = ¢. By (c), there exists w1 such that w1 [w 1 thus w1 [w by (a). Again, w1 j F 1 U F 2 , thus there exists w2 such that w2 n (F 1 uF 2 ) = ¢ and w2 [w1• By induction, there exists then a sequence w'n (n E-lN) such that w'n+ 1 [w'n and

w'n n [F 1 u ••• uFn] =¢for each n E-IN. By (d) 00

¢

I= n

n=1

w'n

c

w

•

However, 00

00

( n WI) n ( u Fn) n n=1 n=1

¢

,

hence a contradiction. Proof of prop. I.1.5. If X is complete and metrizable, let d be a distance compatible with the topology of X; define diam e = inf {1, 2

sup d(x,y)} x,yE-e

•

Then the relation w [w' ~> ~

< w'

and diam w ~

I

diam w'

verifies the above conditions. For {c), observe that


w

< w'

and w relatively compact.

DEFINITION I.1.7. Let E be a l.c.s. A barrel in E is a clo00

u ne. The space n=1 E is barrelled if every barrel of E is a neighborhood of zero. sed absolutely convex subset e such that E =

COROLLARY I.1.8. Every U-space is barrelled. Indeed, let E be an U-space and 9 a barrel in E. We have 00

u ne. Moreover E is a Baire space and ne is closed for n=1 0 each n. Thus ne ~ ¢ for some n. Since e is absolutely convex,

E 0

9 is also absolutely convex, thus it contains O, hence the conclusion. Some deeper properties of meagre sets will be developped in chap.VII, §3.

3

2. THE CLOSED GRAPH THEOREM FOR PROPOSITION I.2.1. E

~

~

E, F

~-SPACES.

~~-spaces

and T a linear map of

F. If the graph of T is closed in E x F, then T

~

continuous. Denote by Ui (i ~:m) and Vi (i ~lN) a countable base of neighborhoods of 0 in E and in F. We may assume that ui+1 + ui+1 c ui and vi+1 + vi+1 each vi is closed.

c

vi for each i ~ lN and that

Step a. For each k we have 00

E

= T -1 F = u

n=1

n

T- 1 V

k

00

C

u n=1

n

T

-1

(1)

Vk • -1

Since E is a Baire space, by prop.I.1.4, some n T Vk has an interior point hence, by the argument of cor.I.1.8, it is a neighborhood of zero : there exists ik ~ :m such that

Comment. We can read (1) as follows : take ~bsolutely convex neighborhood of zero in F, say V. Then T- 1 v is a barrel in E. Thus the assumption "E barrelled" would suffice for ste'P a. Instead of doing so, we have implicitly reproduced the proof that a Baire space is barrelled. But then, by prop.I.1.2, we have some more information on T- 1 vk : it is non meagre. This is of no use in this context but will be essential later on. Step b. Assume that, using ste'P a, we have determined an increasing sequence ik such that

(2) The point now is to get rid of the closure in the right-hand side. Indeed, without closure, (2) would mean that T is con-

4

tinuous. It follows from (2) that U.

~k

C T

Take xk

-1

V k + U.

~k+1

1fk •

• There exist yk

E- U.

~k

0

,

0 0

such that

So we can find inductively a sequence xk (k>k 0

)

and yk(k~k 0 )

such that E-

u.

k

~k

0

Summing up both sides of xk we get

thus

On the other hand, n n T( I: yk) = I: k=k 0 k=k

Tyk 0

Since Tyk E- vk for each k, the series

Tyk is a Cauchy

I: k~k

0

sequence hence, since F is complete, it converges to some point z. Moreover, n I:

k=k 0

Tyk E-

n I:

k=k 0

'In

'

thus z E- vk -1" 0

Now, since the graph of T is closed in E x F,

5

n I:

k=k 0

Yk -

xk 0

n

T ( I:

k=k

0

was an arbitrary element of U.

Since xk

, we have thus

l.k

0

0

proved that

hence the conclusion. Comment. The metrizability of E and F was strongly used through the existence of countable bases of neighborhoods of zero in both spaces. The completion of F was used for the convergence of I: Tyk' k

It is however not difficult to improve prop.I.2.1 in two ways. PROPOSITION I.2.2. Prop.I.2.1 holds true when E is a barrelled space. Take as above a countable base V. (i E-lN) of neighborhoods of l.

zero in F. We have already observed that step a holds true when E is barrelled. It provides a sequence Ui (i E-lN) of neighborhoods of zero in E such that U.l. C T

-1

V.l. , Vi E-lN •

It is of course not a restriction to assume that Ui+ 1 + Ui+ 1 CUi for each i f lN. However, the U. 1 s no longer form a base of l.

neighborhoods of zero in E. As in step b, starting from xk

f 0

Uk , we determine 0

xk+ 1 ,yk (k~k 0 ) such that xk = yk + xk+ 1 , xk f 6

Uk' TykE- Vk, Vk

~ k0

•

The series

Tyk still converges to z

I: k~k

E-

0

vk -1 and, moreover, 0

n I:

xk

k=k 0

0

yk +

X

(1)

n+ 1' Vn •

Let U be an arbitrary neighborhood of zero in E and fix n f IN • By ( 1 ) , n

+ T

E yk k=k 0 + T

-1

Vn+1 +

-1

V

n+ 1

u •

Hence there exists z 1 such that n

E yk - z 1 k=k 0 and Tz' n

E-

E

k=k 0

E-

x

ko

( 2)

+ U

v n+ 1 • On the other hand, Ty,_ 1\.

-

00

z =

E

k=n+1

Tyk

E-

v n+1 + v n+2 + •••

c

v n"

Thus

( 3) Combining (2) anc (3), we see that (xk ,z) is in the closure 0 of G(T) in E x F, hence Txk = z, which yields thE conclusion. 0

There is another interesting proof of prop.I.2.2, based on duality arguments. DEFINITION I.2.3. If T is a linear map of E into F, the transpose T* of T is the linear map of F* into E* (algebraic duals of F and E respectively) defined by <x,T*y'>

= F'

~

8F

8

Prop. II.3.2 and II.3.4 remain true for

F·

The proofs are entirely similar. However, the analogue of prop.

II.~.10

for

f>F

is unknown.

It is essentially because the arguments using the closure no longer hold. There is a partial converse to prop.

II.~.10,

due to

Iyahen [60]. However, we quote it here under the additional assumption that F is complete which seems necessary in its proof. PROPOSITION II.~.1~ [60). ~ E be a l.c.s., La subsuace of E of finite codimension and F a complete l.c.s. 1L L belongs ~

8F' ~ E belongs to &F. We may assume that L is of codimension one. Let T be a linear map of E into F such that G(T) is closed

in Ex F. Its restriction TIL to L has a closed graph in L x F, so it is continuous. Assume that L is dense in E. If x 0 f

E, the filter of

neighborhoods of x 0 in E induces a Cauchy filter

~

in L. By

the continuity of TIL' TIL~ is thus a Cauchy filter in F. Thus it converges to some y 0 f F. Since G(T) is closed, (x 0 ,y 0

f

)

( x 0 , Tx 0

)

..,..,..,..,......,..,ExF C G(T) and Tx = y 0 • Thus G(TIL) 0 ExF ExF -::-Gr.(T:"1~-L~) f ~Gr.(T:"11r-L"T) and G(T)

On the other hand, TIL has a unique continuous extension ~~~ExF

T 1 toE, with graph G(TIL) • Thus T = T• and Tis continuous. If L is closed in E, E is isomorphic to L x JR (or

1}

+ p,

1/U E- U

•

Thus X

fp(Tx)

x

e- u,

p(Tx)

> 1Jue-u

is a basis of a filter converging to zero in E. Its image by T is a basis of a filter contained in the compact set {y: p(y) = 1}. Soithasan adherent pointy. Then (O,y) E- 'G"('T), but y

=/= o, thus ( O,y) f G(T). Hence the first assertion.

*For the second one, we may assume without restriction that F = ~. A linear map of E into F is thus a linear form. If its graph is fast sequentially closed, for every Banach disk B of E, the graph of TIEB is closed in EB~ (since EB and F are Banach spaces, EB x F is a Banach space). Thus TIEB is continuous of EB into F. Therefore T is a fast bounded linear form on E (cf. def.III.2.6). So, in the space E, every

21

fast bounded linear form must be continuous. By prop.III.2.7, it is equivalent to say that E}c is complete. PROPOSITION II.3.16. l !

&u

~ &~

u

is the class of all

l.c.s.,~u,

~u,

are the class of all l.c.s. equipped with their fi-

nest locally convex topology. Let E belong to &U and let t be the finest locally convex topology on E. The identity map of E into Et has a closed graph, thus it is continuous, and E = Et. Conversely, equip E with its finest locally convex topology t. If R is a linear relation of E into a l.c.s. F such that ~(R)

= E, let tR be the coarsest locally convex topology on E

which makes R continuous : if U is a base of neighborhoods of zero in F, {R- 1 (U) U E- u} is a base of neighborhoods of zero in E for tR. Since t is finer than tR' R is continuous, hence

E E-

~R.

Same argument for the other cases. REMARK. In prop. II.3.16, we oan obviously reduce U to the class of all l.c.s. equipped with the finest locally convex

to-

pology. Hence, for instance, prop. II.3.16 is still true when U is the class of barrelled spaces, ultrabornological spaces, etc. PROPOSITION Ilo3o17o [Mahowald, 7 ] . j ! 'iFis the class of all Banach spaces, led spaces.

~U ~

&u are equal to the class of all barrel-

Every barrelled space belongs to ~U , hence to &U • Indeed, let E be barrelled and R be a linear relation of E into a Banach space F such that ~(R)

E

X

= E and G(R) is closed in

F.

For every x E- E, if xRy, then xRy 1 ~ y

1

E- y + R(O)

•

Moreover, R(O) is a vector subspace of F and it is closed, since

22

{0}

R(O) = G(R) n ( {0} x R)

X

•

Thus R induces a linear map T of E into F/R(O), where F/R(O) is another Banach space, and G(T) has a closed graph, since G(T)

{0}

with

=

I ( {0}

G(R)

x R(O)

R(O))

X

'

C G(R).

By prop.I.2.3, T is thus continuous. Since, for every e C F, R- 1 (e) = T- 1 being the image of e in F/R(O)), R is thus

e (e

continuous. Conversely,

&~

is contained in the class of all barrelled

spaces. Let E belong to

Let E 9 be the space E equipped with the topology defined by the semi-norm p 9 , gauge &~and

9 be a barrel in E.

E9

of e, and

the completion of E 9 /ker p 9 • It is a Banach space. The canonical map j of E into E9 has a closed graph. Indeed, assume that (x,y) f

E•

of

Given co

> o,

G(j). Denote by

choose x' f

9

the unit ball

E such that jx' - y f

For each neighborhood U of zero in E, there exists z f

...

-

co

e.

E such

-

u and jz y f e:oe. Then jz jx' f 2c 9, 0 2e: oe 2 e: 0 e and (x-x 1 +U) n 2 e: 0 e :1= p. Thus X - x' f 2 e: 0 e and jx f jx' + 2 E S( y + 3e: 0 Therefrom jx = y and j has a clo0 sed granh. that x-z f

z

-

x' f

e.

By assumption, j

is thus continuous, hence 9 is a neighbor-

hood of zero in E and E is barrelled. *PROPOSITION II.3.18 [18]. ces,

~~and

&~are

1£

~is

the class of all Banach spa-

equal to the class of all ultrabornological

spaces.

We know that ~~ C &~. By prop. 11.2.4, ~~ contains all Banach spaces. By prop. 11.3.12, it also contains all ultrabornological spaces. Assume now that the l.c.s. E belongs to

&~.

Let 9 be an ul-

trabornivorous disk of E (cf. def. 111.2.1). Define the Banach space

E8

and the map j

as in the preceding proof. To see that 9 23

is a neighborhood of zero, we have to verify that j is c ontinuous or, since E E- gt that G( j) is fast sequentially closed. fF ' I f (xm' jx ) is fast convergent to (x,y), there is a fast m compact disk K such that xm- x in EK (prop. III.1.7)• Since 9 absorbs K, it follows that x m x for p 9 • Thus jx m - jx in E. . But jxm - y in E • Thus y = jx and G(j) is fast sequentially 9

closed. 4. THE RANGE SPACE DEFINITION II.4.1. Let us denote by

~E

the class of all spaces

F such that every linear relation R of E into F such that ~(R)

= E and G(R)

is closed in E x F is continuous. If f1 is a

class of l.c.s., denote by ~g the intersection of ~E (Ef&). The next two natural questions are a) to find good permanence properties for

~E

and

~g•

b) to characterize ~g for important classes &. The following results about question a) are essentially due to Raikov [87]. PROPOSITION II.4.2.

~

F, G be l.c.s. and T a continuous li-

near map of F ~ G. If F E- ~E' ~ G E- ~E. Let R be a linear relation of E into G. Then T- 1 o R is a linear relation of E into F. If the graph of R is closed in E x G, the graph of T

-1

oR is closed in E x G

let x , y be a a generalized sequences such that x - x in E, y - y in F and a a x (T- 1 oR)y (i.e. xaR(Ty )). Since Tis continuous, Ty - Ty, a a a a 1 thus xR(Ty) and x(T- oR)y. IfF E- ~E' T- 1 oR is thus continuous, hence R = ToT- 1 oR is continuous. COROLLARY II.4.3. 1£ Ft E- ~E' ~ Ft 1 E- ~E for every locally convex topology t' ~ t. 1£ F E- ~E , then F/N f ~E for every closed subspace N s£ F.

24

PROPOSITION II.4.4. l£ F f F belongs to

~E

, then every closed subspace G

~

~E.

Let R be a linear relation of E into G. It is of course a linear relation of E into F. Since E x G is closed in E x F, the graph of R is closed in E x F if it is closed in E x G. Finally, R is continuous of E into F if and only if it is continuous of

E into G. Hence the result. DEFINITION II.4.5.

Let

~

0

be the class of l.c.s. F such that

every linear relation of E into F such that

~(R)

is non meagre

in E and G(R) closed in E x F, is continuous of E into F and such that ~(R) = E. The class

~0

contains all

~-spaces.

It will be proved in a

more general context in prop. IV.5.3. It is clear that the above results (prop.II.4.2-3-4) hold true for

~

0•

PROPOSITION II.4.6. ( i E-m) Eo ~0 ~ F. ~

l£ '

F is the union of countably many subspa-

~

F Eo ~o·

Let R be a linear relation of E into F, such that ~(R) is 00 R- 1 F. is non meagre non meagre in E. Since ~(R) = u R- 1 F. ~ ~ ' i= 1 for some i E-JN. Consider then the restriction R 1 of R defined by xR'y

~>

xRy and y f

Fi

It is a new linear relation of E into F.

~

non meagre in E and G(R

1 )

, such that ~(R 1 ) is

is closed in E x F .• It is thus conti~

nuous. This yields the continuity of R for every nei~hborhood 1 1 V of zero in F, R- v ) R 1 - (vnF.) and R•- 1 (VnF.) is a neighbor~

~

hood of zero in E. Moreover ~(R) ) ~(R') =E. Hence the result. COROLLARY II.4.7. A countabl• inductive limit of elements of belongs to ~ •

~

0

Indeed, if F is the inductive limit of the spaces F. (i f:m), ~

it is the union of the images (o,y) f l>

y =

To

'G""['T)

= G(T)

= o.

It is clear that t" ~ t. Thus t~ = t& ~ t. On the other hand, Tis continuous o£ (F,t 1 ) into (E,t"), hence o£ (F,t 1 ) into (E,tg)' by prop. II.4.9. Hence the result. A dual.characterization of Br(&) is easily obtained when & consists of Mackey spaces.

27

PROPOSITION II.4.12. Assume that & is contained in the class of all Mackey spaces. ~ (E,t) f

Br(&) if and only if every

vector subspace L ~ E* such that L n E 1 is a(E•,E)-dense in E 1 and that (E,~(E,L)) f

&, contains E'.

Here ~(E,L) denotes the topology of uniform convergence on 0

(L,E)-compact absolutely convex subsets of L. The condition is necessary. Take such a subspace L and consi-

der the identical map i that ~(i*) = L n E

1 •

of (E,~(E,L)) into (E,t). It is clear

So the density of L n E 1 in E' means, by s

prop. I.2.5., that i has a closed graph. It is thus continuous, hence ~(i*)

=

E 1 and L) E 1 •

The condition is sufficient. Let t' be coarser than t. The identical map i

of (E,t') into (E,t) has a closed graph. Thus,

by prop. I.2.5,

(Et

1 ) 1

g Et' is a Mackey space,

nEt is dense in E~. Moreover, since

g

(E,~(E,(Et 1 ) 1 )) g

Thus (Et' ) 1 g

)

=

(E,tg) E & •

E 1 and, since Et' is a Mackey space, t g

0~

t. The

conclusion follows by prop. II.4.11. We refer to the literature for further developments along this line. We have sketched above an approach to a characterization of the classes of range spaces. The descriptions of these classes turn out to be difficult to handle for practical purpose, i.e. when deciding, in concrete examples, wether a space belongs to them or not. Thus it may be more important to exhibit nice subclasses of ~&' large enough to contain those spaces that occur in practice and having reasonable permanence properties. We already mentionned that the set & of all ultrabornological

-

spaces would be an acceptable class of domain spaces. There do exist interesting subclasses of the corresponding

~g•

The first

one was introduced by Raikov [87]. Next L. Schwartz showed that contains all Souslin spaces. Finally, the author introduced the class of webbed spaces. The theory of these spaces will be

~g

developped in chap.IV,V,VI and the results of L. Schwartz in chap.Vn. 28

III Bomological and ultrabomological spaces

1. FAST CONVERGENT SEQUENCES To substantiate the idea that the class of ultrabornological spaces is adequatly comprehensive for the needs of applications, this chapter is devoted to their study. The strong analogy with bornological spaces makes it reasonable to handle both cases together. DEFINITION III.1.1.

Let E be a locally convex Hausdorff space.

An absolutely convex subset. A of E is also called a disk. If A is a disk of E, tA denotes the topology for which

{x + r.A : r. ) 0}

is a base of neighborhoods of x for ePch x E E. The linear hull of A equipped with the topology induced by tA is a semi-normed space, its topology being defined by the semi-norm pA , gauge of A. We denote it by EA.

If EA is normed, we say that A is

normin~

If it is moreover complete, A is a Banach disk (or a completing ~).

A disk A absorbs a subset e of E if e C AA for some A. LEMMA III.1.2. The disk A absorbs e if and only if, given a seguence An E- ]0,1] (nE-lN), A absorbs sequence x n

If A does not absorb e, r

n E lN } .:.f~o::.;r.._.:::.e~v..::e:..:r~yL.

{A n x n

(n E- lN) E- e. take xn in e "- 2

n

-1

An A. Then, for each

E-lN, Ar xr ~ 2rA, thus A does not absorb {Anxn : n E-lN}.

The converse is obvious. DEFINITION III.1.3. A sequence x (n flN) is Mackey convergent n

(reap. fast convergent) to x if there is a bounded disk (reap. a bounded Banach disk) B of E sue~ that xn -

x in EB. A Mackey

(or fast) convergent subsequence is clearly convergent to the

same limit. A subset K of E is fast compact if it is compact in EB for some Banach disk B. PROPOSITION III~1.4. .:.C:..:O~n::..v~e~x~h::..u=l~l:........:o~f:..

If x (n ~ m)- O, the closed absolutely n n ~ nq is the set of all convergent series

{xn

~ I en I :!!: 1. If all such series are n=1 n=1 convergent, co({xn: n ~ m}) is moreover compact and metrizable. 00

~

of the form

00

en xn .!!i.1h

Conversely, i£ E is metrizable, every precompact subset of E is contained in the closed absolutely convex hull of a sequence xn(n~m),

convergent to zero. 1\'!oreover, the sequence xn(n ~ JN)

can be choosen in a dense subspace of E. Consider the linear map T of 1 1 into the completion defined by Tc(n~m) n

~

C

n=1

X

n

E of

E

n

It is continuous from the unit ball B of 1 1 , equipped with the a(l 1 ,c 0

E.

norm p of ~

n=1 Given £ >

sup n:!!:N

en

o,

Indeed, take any continuous semi-

We have

00

p(

E.

topology, into

)

X

n

-

N

00

~

n=1

CI n


N

I en -c~ I < £/2 [

X

n

) :!!:

lcn-c~l p(x n ) + 2 sup p ( xn) •

~

n=1

n>N

£/4 for N large enough, thus

N

00

~ n=1

p(x )+1] n

=>

p( ~

C

n=1

n

X

n

~

n=1

c'n x n )l

•

The sequence xm(m E-lN) is precompact for pB 1 hence [89, p.51 precomp~ct

43, p.200], B 1 is sup x 1 E-{xm:m E-lN}

l<x,x'>l o

or

for sup x 1 E-B 0

l<x,x'>l

and, using once more the same argument, B is precompact for pB,. Since moreover B 1 is closed and absolutely convex in Ea(E,E•)' [43, p.95 or 62, p.395], tB, and a(E,E 1 ) coincide on B. Since y is bounded on B 1 , it is continuous for tB'' hence continuous for a(E,E 1 ) on B. Hence the proof. PROPOSITION III.2.9. The space E is bornological (resp. ultrabornological) if and only if it is a Mackey space and E' me (resp. E£ 0 ) is complete. Prop. III.2.9 and the main ideas in the proof of prop. III.2.8 are due to Kothe in the bornological case[62, 63]. If E is ultrabornological, it is bornological. If it is bornological, since the neighborhoods of zero for the Mackey topology ~ of E are bornivorous (Mackey boundedness theorem), E is a Mackey space. By prop. III.2.8, if E is ultrabornological, E£ 0 is complete, since (Eub)' 38

= E 1 • If E is bornological,

E~ 0 is complete.

Indeed, let

~

be a Cauchy filter in E' • It converges for me 0 (E*,E) to some y' f E*. For every sequence x (m f m) Mackey o m convergent to zero, there exists y' f E' such that sup l<xm,y~-y'>l ~ 1 • m Thus

y~

is locally bounded, hence continuous.

Conversely, if E~c (resp. E}c) is complete, then {Eb)' {resp. (Eub)' {resp. E

= E1

= E 1 ) . If E is moreover a Mackey space, E = Eb

= Eub).

3. PERMANENCE PROPERTIES AND EXAMPLES Let us first mention an easy connection between bornological and ultrabornological spaces. PROPOSITION III.3.1. l£ E is bornological and sequentially complete, it is ultrabornological. Indeed, every bounded disk is contained in its closure which, by prop. III.1.9, is completing. PROPOSITION III.3.2.

~

T be a linear map of E

~

F.

1£

E

is bornological (ultrabornological) and if T is open and maps Mackey convergent {resp. fast convergent) sequences into bounded sequences, ~ F is bornological {resp. ultrabornological). If A is a bornivorous (resp. ultrabornivorous) disk of F, 1 T- A is bornivorous (resp. ultrabornivorous) in E (for the latter case, apply prop. III.1.11). It is thus a neighborhood of zero in E, hence A is a neighborhood of zero in F. PROPOSITION III.3.3. The inductive limit of bornological (ultrabornological) spaces is again bornological (ultrabornological). Let E be the inductive limit of the spaces E and T be the canonical map of E to E. If A is bornivorous i~ E, T-~A is a a bornivorous in E • If E is bornological, it is thus a neigha

borhood of zero in E and this for all a implies that A is a a neighborhood of zero in E.

39

Same proof for the ultrabornological case. REMARK. Since it is obvious that finite dimensional l.c.s. are bornological and ultrabornological, we have thus proved that the class of all bornological (resp. ultrabornological) is inductive (cf. def.II.4.8). PROPOSITION III.3.4. ~ F be a subspace of (E,t) such that every locally (resp. ~) bounded linear form on E vanishing ~ F is identically zero. l£ F is bornological (resp. ultrabornological),so is E. By use of the Hahn-Banach theorem, F is dense in Eb (resp. Eub). By prop. II.4.9, we have, in F,

By assumption tb(F) = t on F, thus tb(E) = t on F and, due to the density of F in (E,tb), tb = t on E. Same argument for tub• PROPOSITION III.3.5. The space mX (~eX) is bornological and ultrabornological if card X is smaller than the first strongly inaccessible cardinal. A cardinal a is said to be strongly inaccessible if (i) a (ii) ~ (iii)

>
= 1 and denote by T the projection

x - x - <x,x'>x 0

of E onto L.

Assume first that x'

is locally bounded. Then T maps bounded

sets of E into bounded sets of L, If 9 is a bornivorous disk of L, T- 1 9 is a bornivorous disk of E and 9 = L n T- 1 e. If 9 1 is a bornivorous disk of E, 9

1

n

L is bornivorous in L. Hence Lb has If E is bornological,

the topology induced by Eb.

L is thus bor-

nological too. Assume now that x' bounded sequence x

~

X

m

is not locally bounded. There is then a

m such that <x ~ ,x'> = m for each m. From

1

m -= -m Tx m + x o 1 - Tx m m

we deduce that Ym

X

the absolutely convex hull of

X

0

in the space EB , where B 0 is

0

and x ( m E- lN), m

0

thus a bounded

disk of E. Let 9 be a bornivorous disk of L. Denote by

~

the family

all bounded subsets of E containing B • For each B exists CB > 0 such that CB B

n

L C

e.

0

~

~.

of

there

Define 9

=

1

C~B)

Co( U

•

BE-~

It is of course

a

bornivorous disk of E. Moreover,

e' n 1 c 2e • Indeed, if

X

X

E- 9

1

( *)

n

L, we have

N !:

N !:

i=1

i=1

N

ai T z i

+

E i=1

a 1.

(y.m -x o )

~ CB for m large enough, hence

1

B.1. + ~ (z.l. ,x 1 )B o C 2CB. Bi C 29 m 1

(*).

Take now 1

9 " = c o ( 9 u 29 ' ) ;

n

it is a bornivorous disk of E, such that 9"

L =

e.

It is obvious that a bornivorous disk of E induces a bornivorous disk of L, thus we have proved that Lb has the topology induced by E 0 • This implies as above that L is bornological if E is bornological. See [110] for the ultrabornological case. Let us now turn to criteria for dual spaces to be bornological. As far as applications are concerned, the main result is the following, due to L. Schwartz [97] when E is a Schwartz space and improved in the present form by Hogbe-Nlend [48]. DEFINITION III.3.9. The space E is called an infra-Schwartz space if, for every neighborhood U of zero, there is an absolutely convex neighborhood V of zero, such that U0 is weakly compact in the Banach space generated by

vo.

PROPOSITION III.3.i0. The strong dual of a complete infraSchwartz space is ultrabornological. Let E be an infra-Schwartz complete space. Denote by b(E•,E) the strong topology of E 1 and by t the topology of the inductive limit of the Banach spaces E

00

(U neighborhood of zero in E).

Since every U0 is bounded for b(E 1 ,E), it is clear that t ~ b (E 1 ,E). Let us prove that (Et)' =E. This will imply-that t i s coarser than the Mackey topology of E 1

,

hence coarser than

b(E•,E). Therefore Eb(E•,E) will be equal to Et' hence ultr~­ bornological. Since E is complete, by Grothendieck's completion theorem, a linear form on E' belongs to E (up to obvious identifications) if and only if it is continuous on each equicontinuous subset

43

of E' for the topology a(E•,E). Assume that x' ~ (Et)'. Given a neighborhood U of zero, there is another one V such that U 0 is weakly compact in Evo• The form x' is bounded over vo, hence continuous in Evo for the norm of Evo• It is thus continuous for the weak topology of Evo• Since uo is weakly compact in Evo' the weak topology of Evo is equal on uo to a(E•,E). Thus x' is a(E•,E)-continuous on U0

Hence the conclusion.

•

A converse to prop. III.3.10, identifying every ultrabornological space to the strong dual of a complete nuclear space, can be found in [48]. The following results are due to Grothendieck. DEFINITION III.3.11. A space E is called evaluable

if every

closed bornivorous disk is a neighborhood of zero. PROPOSITION III.3.12.

liE

is metrizable, the strong dual of

E is ultrabornological if and only if it is evaluable. First, if a space is ultrabornological, it is of course evaluable.

m)

Let Ui(i ~

be a base of absolutely convex neighborhoods

>

of zero of E. For every sequence A. in

Eb

~

0 (i ~

m), the closure

of co

9 =Co(

U Ai U~) i=1 ~

(*)

and its algebraic closure are equal. Recall that the algebraic closure

A of

a disk A is

n£ >0 (1+c)A.

It is clear that

To prove the converse, observe that, since each N

Di

a c 9.

is absolute-

ly convex and a(E•,E)-compact, Co( U A. U~) is also a(E 1 ,E)i= 1 ~ ~ compact [43, p.91], thus

N co( u

u~) ~ ~

A·

i=1

Take x' each N,

44

f

a.

N

= ( n .L Ai i=1

U.; ...

)o

There is some £ )

•

0 such that x'

f

(1+£)9. For

m)

sequence xn(n ~

N L

n ui such that <xN,x'> > 1+E. The i= 1 A.i is bounded. If w is its polar in E•, we have

hence there exists XN f

(x 1 +Ew) n 9 = ¢ • Indeed, otherwise there would exist N f N

u

Co(

y' f

i=1 such that x'

-

m

and

N

= ( n L u.l. ) 0 A.i U'?) l. i=1 A.i

y'

EW• How ever

~

hence a contradiction.

Eb

Assume now that nivorous disk in E

1 •

is evaluable and that 9

Since each U'? is a Banach disk, 9' conJ.

tains some 9 of the form (*). Then 29 1

)

e=e

e is a closed and bornivorous disk in

Since

bounded subset of

Eb

is an ultrabor-

1

is equicontinuous),

Eb

a is

(closure in

Eb).

(because every a neighborhood

of zero, hence the conclusion.

l!

E is metrizable and Eb separable, then is ultrabornological. By the proof of prop.III.3.12, we only need to check that

PROPOSITION III.3.13.

Eb

co

9

=

Co ( U A.. U'?) i=1 l. l.

(closure in Eb; A.i > 0) is a neighborhood of zero. Since Eb is separable, E' \ 9 is also separable for the topology of

Eb

(because it is open). Let x~(m ~m) be a countable

dense subset of E 1 \9 • For every m, x' 1 e, hence m 1 m m X 1 1 ( n -- U. ) 0 and there exists X ~ n l_ Ul.. SUCh that m i=1 "-i J. m i=1 "-i (x ,x'> > 1. The sequence x (m ~IN) is bounded in E. If w is . m m m 1 1 J.ts polar, 2w C 9. Indeed, if there exists x' E (2w)\ 9, by density, there exists i such that

45

supj<xm,x'-xi>J < 2 m

Then < J<xi,xi_>J <sup

J<xm,x'-xj_>J +sup

m

J<xm,x'>l < 1

m

hence a contradiction. COROLLARY III.3.14.

1£

E is metrizable, separable and semi-

reflexive, Eb is ultrabornological. ~

It is true in particular

E is a metrizable Schwartz space.

Indeed, by definition of semi-reflexivity, every bounded subset of E is contained in a weakly compact subset. Hence E b' -- E ,I · Since E is separable, E' is separable, hence E' is separas 't ble. Apply then prop.III.3.13.

One further step in the study of conditions for Eb to be bornological or ultrabornological consists in considering the dual of an inductive limit. The following result slightly extends a theorem of Grothendieck [6]. PROPOSITION III.3.15.

~

E be the strict inductive limit of a

sequence of evaluable spaces En (n ~ m). If each E'n, b is bornological, Eb is ultrabornological. Moreover, every bounded subset of E is contained in the closure of a bounded subset of ~ En•

REMARK. Since E n is evaluable, every bounded subset of E'n,b is equicontinuous, hence its closure is ~omplete. Thus E~ is quasicomplete and it is simultaneously bornological and ultrabornological. The same is true for E•. Denote by b 1 the topology of uniform convergence on the bounded subsets of the various En •s. We shall first prove that Eb' is bornological. Let

46

e

be a bornivorous disk in Eb,. There exists

~lN

such that

E~

E- e.

Otherwise, for each n, choose

x~

E- E~ \ e

0

hence

vanishing on E n • It is obvious that the sequence nx'n converges to zero in Eb'" Thus it is a bounded subset of Eb' and it is absorbed by e there exists C > 0 such that n

X I

n

lfn E- JN

€- C e t

which contradicts x~ Denote by en

f

t

e for n

> c.

the set of all linear forms on En 0

which can 0

be extended to E by an element of e. It is a bornivorous disk of E' b" Indeed, if B is bounded in E' b' since En is evano' no' o luable, it is equicontinuous. Since the inductive limit is strict, it is also the restriction to E

n

subset B 1 of E

For some C

1 •

> o,

of an equicontinuous 0

B 1 C C e. Thus B

c c en

0

The spaces E'n, b are assumed to be bornological. Hence b : there is some bounded en is a neighborhood of zero in E' o n no' 0 0 (polar in E~ ). set B in En such that en ) B 0

0

If now B 0 is the polar of Bin E 1 restricted to En 0

0 ,

for every x' E- B 0

coincides with the restriction to E n

,

x' of an

0

element of e. Thus B° C e + E° n

hence

t~e

C 2e 0

conclusion.

It is obvious that the topology b is finer than b 1 •

Let us

prove that they have the same bounded sets. It will imply that the absolutely convex neighborhoods of zero in Eb are bornivorous in Eb'' hence are neighborhoods of zero in Eb' and therefore b' = b. If B is bounded in Eb'' its restriction to En is bounded in E'n, b' thus equicontinuous for each n E-m. Therefore B is equicontinuous onE hence bounded in E'. Let finally B be bounded in E. ~ts polar is a neighborhood of zero in Eb' = Eb. Hence there exists a bounded subset B 1 of some E n such that B 0 ) B 10 Therefrom BC'C'O B', hence the last assertion of prop.I.).16.

47

IV Webbed spaces

1. WEBS DEFINITION IV.1.1. Denote by I the set { ( n 1 , ••• , nk)

k, n 1 , ••• , nk E- IN J

&

ordered by the following order relation a (m 1 , ••• ,mk) ~ (n 1 , ••• ,nk,) Let E be a l•c.s. A

~ ~


v n for vn all n E JN is called a strand of the web. So a strand is a se-

A

sequence e

quence e( ki i A

n1, ••• ,nk.

)(i EIN), where nk(k EJN) is arbitrary and

~

co.

web is said to be completing if each of its strands is

completing. It is strict if it is absolutely convex and if each of its strands is strictly completing. The space E is said to be webbed or strictly webbed if it admits a completing or a strict web. The following easy remarks are quoted for later reference. PROPOSITION IV.1.5.

If a l•c.s. E admits a completing

it

web~,

admits another completing web which elements are starshaped around zero. The new web ::R. 1 constituted by the subsets e'

v

=

u

H[0,1]

Ae

v

(e

v

E::R.)

is obviously completing if :7r. is. PROPOSITION IV.1.6. For each corr.pletin~ (resp. strict)~ ~

E,

there is another one ::Tr.'

such that each subset of E absor-

bed by the elements of a strand of :7r. is contained in the elements of a strand of The new web ::R.'

::rr.•.

is defined as follows

49

e'

n1

= m e (m 1 ,n 1 E-lN) 1 1

n

indexing (m 1 ,n 1 ) be a single index n1 E-m. Next, indexing (m 2 ,n 2 ) by n2 E- IN, e n1 1 ,n , = m1 e n n m2 e n ,n 1 2 1 2 1 and so on. We clearly obtain a new web.

=

A

strand of

~1

reads

n m2e n , n n ••• n mk i e n , ••• , nk. (mk,nk E-m for n1 1 1 2 each k; ki i oo), If A.i(i E-lN) is associated 1 or strictly associated to the strand e (i e-m) of ~, then n 1 , ••• , nk

&i

m,e

i

>..! J.

= A· / J.

sup mk

~k.

J.

is associated (or strictly associated) to the sequence &i (i E-:N), Comment. Completing and strict webs were introduced in [14], "r~seaux

under the name of

de type

~

et

r~seaux

stricts". We

use here the terminology of [89] and [90], although with a slightly different definition. In [14], it is assumed, in the definition of "completing webs", that A.k

.f=o

for each k E-lN. The slight improvment assu-

ming that only a subsequence does not vanish was introduced by M. Nakamura

[76].

Although bringing some technical troubles

later, it does not affect the properties of the webs and turns out to be useful for a topological interpretation of the completing webs. We had- however to change a little bit Nakamura's condition to preserve the stability with respect to products and projective limits. On the other hand, in condition (c) of def.IV.1,1, it would not make any difference to assume that U e is absorbent in v )j.l.

e

ll

v

(instea4 of being equal toe ). Prop.IV.1.5 would then proll

duce another web verifying (c) in its initial form. Various other types of webs are also introduced in [14]. M. Nakamura has proved the equivalence of several of them in

[76]. 50

The next elementary properties of completing sequences will be useful later. PROPOSITION IV.1.7. ~ Bn(nE-lN) be a completing sequence of E, A (n E-JN) an associated sequence of numbers and U a neighborn hood of zero in E. ~ k large enough, 00

{ ~ ~.x. i=k ~ ~

: ~i E- [O,Ai], xi fBi}

In particular, A.iBi C U

£££

~

i

C U •

k.

It is not a restriction to assume that U is closed. So it is enough to prove that n

{ ~ ~i xi : n ~ k, i=k

~i E- [O,A.i], xi f

for k large enough. If it is not the case,

Bi}

C U

there exist, by in-

duction, sequences mk,nk Too, ~i E- [O,Ai] and xi fBi such that nk




b ~

Choose Ai

) 0 ( i E-lN) and n.~ f

~

such that

~i x 1 converi=1 ges for all ~i E- [O,Ai] and xi E- Bn. • We may assume that ~

Ai !

o. Put

53

B!

l.

or

course Bj_+1 n Ak = { E i=1

c

B! for each i E- IN • Define next, for k l. l.

1'

n

co ( i
X.

~

E

i=1

B' • i2k

It is clear that

Moreover, Ak+ 1 + Ak+ 1 C Ak' ~k ~ 1 • Indeed, for each n, n

n

n

n

C E B1 + E B' E B1 + E B1 C A 2i2k k. i2k+ 1 i=1 i=1 i2k+ 1 i=1 (2i-1 )2k i=1 By prop. IV.1.7, for every neighborhood U of zero in E, An C U for n large enough. There is a unique topology t

on E which makes E into a addi-

tive topological group and admits Ak(k

e-m)

as a base of neigh-

borhoods of zero. Moreover, t is finer than the initial topology t

0

of E, Et is metrizable and, for each k, B'k C Ak. 2

Let G be the completion of E with respect to t

it is a me-

tric complete group. We are left to show that the identical map i of Et into E extends to a continuous homomorphism into E. co

~

of G

Given x E- G, there exist x 0 E- E and xn E- An such that co

E xn converges to x in G. The series E xn is also convergent n=o n=o in E for t 0 • Indeed, each xn (n~1) has the form kn xn

'"'

(n)

'"' y ' i=1 i2n

where y(n) E- B' for each i. It is possible to associate to i2n i2n each (i,k) (i E-IN, k~1) an integer 1 = y(i,k) such that y(i,k)
54

zk ~ B~ for each k, the series

E k=1

zk converges to z in E. On

the other hand, for each i, N E

N

E

k=1

zk -

~

n=1

U (B!+B! 1 + ••• +B 1 n)i 1 1+ n

c.1

)

for N large enough. For every neighborhood U of zero in E, by prop. IV.1.7, we have Cn C U for n large enough, hence N E n=o for

X

n =

X

0

N E xn n=1

+

the topology t

X

+

0

Z

Z

X

when N ""• It is obvious that z depends only on x and not on the choice 0

X

of the x~s. Indeed, if x = Exn = Ex~, then E(xn-x~) hence for t •

0 for t,

0

Defining ~(x) = z X , we obtain a extension of ~ to G, which is still a continuous homomorphism. Hence the conclusion. REMARK. When the sequence B (n n

fm)

is strictly completing, the

above proof can be simplified. Let A.k(k ~m) be a strictly associated sequence. By taking a subsequence of Bn(n ~lN), we may assume that A.k

I

0 for each k ~

m.

Moreover, it is not a res-

triction to assume that A.k+ 1 ~ A.k/2 and E A.i Bi C A.k Bk, Vk f i=k+1

m •

It is easy to check that A.k Bk(k flN) is a base of neighborhoods of zero of E for a topology t which makes E into an additive metric group. Moreover, t is finer !han t

0 •

Now Et is complete.

Indeed, if xk f A.k Bk for each k, E xk converges to some x k=1 in (E,t 0 ). Moreover,

N x -

oo

E xk = E xk f k=1 k=N+1

A.N BN' VN •

Thus x is the limit of Exk for the topology t. However, for the converse, the condition

55

co ~

)..k :Sk C :Sk '

k=k +1

lfko E- lN '

0

0

which plays a fundamental role in the theory of strict webs, does not seem to have a nice interpretation in the group topology.

3. SOME EXAMPLES PROPOSITION IV.3.1. Every

~-space

has a strict web.

Let Uk(k E-lN) be a base of absolutely convex and closed neighborhoods of zero in E such that 2Uk+ 1 C Uk for each k E-m. Then (n1, ••• ,nk)- n1 u1 n ••• n nk uk is a strict web

~.

It is first a web, generated by the sets

muk (m,kE-lN) as in example IV.1.2. Moreover, for each strand e(

n1 , ••• , nk.

) (nk E-lN,k. T co), A. = 1/sup nk is a strictly asso1 1 lc::!::k.

1

1

ciated sequence. REMARK. If E is a :Banach space with closed unit ball ( n1 ' • • • 'nk) -

:s,

n1 :S

is still a strict web and any sequence )..i (i E-lN) such that

co ~

i=1

)...

~

1 is associated to any strand of this web.

1

PROPOSITION IV.3.2. The dual E 1 of a metrizable space E, equipped with any locally convex topology for which the eguicontinuous sets are bounded, has a strict web. REMARK. A subset of Eb is bounded if and only if it is equicontinuous. Thus the finest lccally convex topology for which the equicontinuous sets are bounded is the bornological topology associated to Eb. It is also obviously equal to the topology of the indue ti ve limit of the Euo ( U f U), where U is a base of neighborhoods of zero in E.

56

The important fact here is that prop. IV.3.2 is true for all the usual topologies

t~

of E 1

where~

,

C

~b.

Proof of prop.IV.3.2. Let Un (n E-lN) be a base of neighborhoods of zero in E. We have E 1 The web

~

=

U n-=1

uo. n

of E 1 defined by

is a strict web. Indeed, the sequence 2-k(k E-E) is associated to each strand : if xk E- U~

1

for each k

e-m,

E2-kxk converges

to some point in U0 (since the e( ) are absolutely n1 n1 '• • • 'nk convex, there is no use to consider ~k E- [0,2-k]). Comment. At this stage, it already appears that the "size" of the elements of a web can widely vary from a space to another in one case, they are neighborhoods of zero, in the other one, they are compact for a(E 1 ,E). The webs encountered in the previous two examples don't use the full power of the definition : regardless to the great number of indices, the e(

n1 '• • • 'nk

)'s don't differ very much from

one another. The following example is fundamental since it includes the space of distributions of L. Schwartz. It also uses the full power of the definition. PROPOSITION IV.3.3. lL E is the inductive limit of a sequence of metrizable spaces En (n E-lN), its dual, equipped with any lecally convex topology for which the eguicontinuous sets are bounded, has a strict web. For each n E- lN, let

U~n)(k E-lN)

be a base of neighborhoods

of zero of En• For each n,

~ U~n)o = k=1

E1

,

thus

(k)o ) U••• U U ( k, n 1 , ••• , nk E- lN nk is a web. Given a strand, if xk E- e(n tions of the xk's to E.

~

1 ' ••• '

n ) for each k,

the restric-

k

( i)

belong to U ni

0

for k

~

i. Thus

xk(k E-JN) is equicontinuous on each Ei, hence on E. Thus E2

-k

00

E

2

xk converges in E• and -k

\lk

xk E-

k=k +1

0

•

0

4. PERMANENCE PROPERTIES The existence of a completing or of a strict web in a space E has a very good stability with respect to the standard operations on 1. c. s. Consider first subspaces. PROPOSITION IV.4.1.

~

(resp. strict) web in E.

E be a l.c.s.

!h£li, 1£

and~

a completing

L is a fast sequentially clo-

sed vector subspace of E, ~· = {enL

+¢ :

e E- ~~

is a comple-

ting (resp. strict) web of L. (k e-:m) of~, by prop.IV.1.9, we can choose "k the associated sequences >..k(k E-lN) such that the series For each strand e

00

E k=1

Ilk xk (Ilk E- [o,xk], xk E- e

) "k

are fast convergent. I f xk E- L for all k E- lN' they also c onverge in L. Thus >..k(k E-lN) is associated to the strand e n L(k E-lN) "k of the web ~·. We examine next the image by a linear map. PROPOSITION IV.4.2. ~ E be a l.c.s. and~ a completing (resp. strict) web in E.

1£

T is a linear map of E onto a l.c.s. F,

mapping fast convergent sequences of E into bounded subsets of F,

58

.1.!:!JUl T.1t = {Te : e

E Sll}

is a completing {resp. strict) web of F.

It is clear that TS'l is a web of F. Let e

vk

(k ElN) be a strand co

in .9l and .).k(k EJN) an associated sequence such that (Ilk E [O,.).k]' xk E e

vk

I: Ilk xk k=1 ) is fast convergent in E. By prop. III.

I: Ilk Txk is fast convergent in F, k=1 sociated to the strand Te (k EIN) of TS'l. vk If .9l is strict and the .).k 1 s such that I: k=k +1

1.12,

is as-

Ilk xk E

0

we have I:

k=k +1 0

Ilk Txk = T( I: k=k +1 0

Ilk xk) E Te

vk

0

hence TS'l is also strict. COROLLARY IV.4.}. A completing (resp. strict) web of a l.c.s. E is still a completing {reap. strict) web for Et' where t is any locally convex topology coarser than the associated ultrabornological topology. Indeed, bounded Banach disks of E are bounded in Eub' hence in Et. Apply then prop. IV.4.2 to the identical map of E into Et. Using example TII.2.4 and cor.IV.4.3, it would be enough to prove prop.IV.3.2 and prop. IV.3.3 only for the topology a(E•,E~ COROLLARY IV.4.4. If a l•c.s. E is webbed (reap. strictly web~),

so is E/L for each closed vector subspace L ~E. Indeed, the quotient map is continuous.

REMARK. We have seen in prop.IV.1.10 that a completing web .9l is strict if its elements are absolutely convex and closed. It may seem reasonable for pratical purposes to restrict somewhat the generality by considering only such particular webs. It is however too restrictive. A first reason is that prop.IV.4.2 would fail, unless very special assumptions on T are stated, since continuous images of closed sets are usually not closed. The

59

next result provides important examples of strict webs which elements are not closed. If a 1 • c • s • E ,o~:i;.::s:.,._t.::,h~e~u~n=i~o:,:;n~~o;:;,f_...::c~o:::.u=no.lit..:iia~b:::.:.lo~.Y....::m:::a~n~y

PR OP OS IT I ON IV • 4 • 5 •

subspaces Ei (i ~lN) such that each Ei is webbed (resp. strictly webbed), ~ E is webbed (resp. strictly webbed). I f 9i ( i) =

{e ( i )

v of each Ei, then

e

n1

- E -

:

~

v

I}

is a c omp 1 e t i ng (res p. strict) web

• e( ) n1 ' n1 '• • • 'nk

is a web in E.

( n1) e(

~' • • • 'nk

)(k,n 1 , ••• ,nk ~lN)

It is completing (reap. strict) since, excepting

their first element,

its strands are strands of some 9i(i)•

COROLLARY IV.4.6. A countable inductive limit of webbed (resp. strictlv webbed) sp~ces is webbed (reap. strictly webbed). Let E be the inductive limit nf the webbed spaces E.(i ~~). l.

If t i

~

is the topology of E and ti the topology of Ei, for each lN,

t

is coarser than ti in Ei. Thus Ei is webbed for the

topology t

(cor.IV.4.}).

The conclusion follows from prop.IV.4.5. Same argument for strict webs. For instance countable inductive limits of strict webs. However, except for

£~-spaces,

~-spaces

have

the elements of the

webs are generally not closed. It does not make any difference if, instead of taking vector subspaces Ei of E, we consider the more general situation of an inductive system (E. ,j. ). Simply, use prop.IV.4.2 instead of l.

l.

c or. IV. 4. } • PROPOSITION IVo4o7• .!:&1 Ei(i ~lN) be a seauence of webbed (reap. strictly webbed) l.c.s. and jik(i..~1)(kflN)

be associated to the

f~rst

column,

>..~ 2 )(kE-lN)

Let

to the

second, and so on. Assume for a while that none of them is vanishing. Then

(1)

11 1 = >.. 1

(1)

, 11 2 .. inf ( >.. 2

,

(2)

>.. 1

.

(1)

, 11 ~ = J.nf ( >..~

is associated to the strand of fR.. Indeed, if xk f

e

(2) (~\. , >.. 2 , A.1 '), • ••

vk

, for k

~

i,

g 1 (xk) belongs to the strand corresponding to the ith column and, since the 11 ks are less than or equal to the corresponding asso-

61

00

ciated numbers,

~~ gi(xk)

E

converges in Ei for all

k=1 DO

~~ ~

[o,~k].

E

Thus

~k xk converges in the projective limit

E.

k=1

Now, if the

1~ 1 ) .t{

are not

all strictly positive, take a sub-

sequence of non vanishing terms. The selected indices fix a subsequence of the second strand, thus a new strand in E 2 • Take t..~ 2 )(k ~lN) associated to it and repeat the argument. The end of the proof is then the same as above. RENARK.

It is essential in this proof that a subsequence of a

strand be another strand.

Our definition of completing webs in

[14] assumes that each strand of the form e(

n 1 , ••• , nk

)(k ~JN)

has an associated sequence of strictly positive numbers.

In or-

der to get the topological interpretation of the webs of

§2,

it

is necessary to allow these numbers to cancel. To preserve prop. IV.4.7, we are then obliged to take strands of the form e

n 1 , ••• ,nk.

(i ~lN), with k 1 t

DO•

l.

PROPOSITION IV.4.8. Every countable product of webbed (resp. strictly webbed) spaces is webbed (resp. strictly webbed). Let Ei(i E-lN) be l.c:s. with completing or strict webs .9l(i)• DO

n

The product

Ei is the canonical projective limit of the fi-

i=1 N

nite products

n

E .• Thus, by prop. IV.4.7, it suffices to coni= 1 l. sider finite prodncts. By induct:Lon, it is even sufficient to

deal with the product of two spaces. The same proof would work for the general case but it is much simpler to write it down for two spaces. Define on E 1 x E 2 the web X

(2) e(n, '• • • ,nk) '

indexing each (mk,nk) by a single index.

62

A strand is obtained by fixing sequences mk' nk (k E-lN) and k 1 f ~ 1 denote it ei

=

e( 1 ) x vi

e(~) vi

with

e( 1 ) .. e( 1 ) vi

(m,, ••• ,mki)

; e( 2 ) = e( 2 )

vi

(n,, ••• ,nki

)(ie-m) •

Let A,.(iE-JN) be a sequence associated to e( 1 )(iE-lN). Let 1 vi iJ.(j E-lN) be a subsequence such that A. • .:/= 0 for all j . Then 1j

e(~) v.

is a strand in 9l( 2 ). I f IJ.J.(j E-JN) is associated to it, the

1j

sequence £i (i E-lN) where £i

0 if A.i

=

0 and

£. 1.

=

inf

J

is associated to ei(i E-lN). PROPOSITION IV.4.9. Every countable direct sum of webbed (resp. strictly webbed) spaces is webbed (resp. strictly webbed). It is the inductive limit of the finite products of the spaces. Hence the conclusion, by prop. IV.4.8 and cor.IV.4.6. 5. CLOSED GRAPH THEOREMS Let us start with the simplest case, in order to clear the proof of all cumbersome technicalities. PROPOSITION IV.5.1.

~

E,F be locoso and T be a linear map of

E into F. Assume that E is an ~

~-space

, F admits a strict web

G(T) is fast sequentially closed in E x F. ~ T is con-

tinuous. Let Uk (k E-lN) be a base of neighborhoods of zero in E such that 2Uk+ 1 C Uk for each k E-lN and let 9l = {ev 1 v E- I} be a strict web of F. We essentially duplicate the proof of prop. I.2.1.

Step a. Since

= T- 1F

E

T- 1 e

u n 1 f IN

n1

by prop. 1.1.2, there exists n 1 f

m

such that T- 1

is non

meagre in E. From U

T

-1

n 2 f lN

e(

n 1 ' n2

, , 1

again by prop.I.1.2, there exists n 2 such that T

-1

e(

non meagre.

n1 'n2

) is

Hence, by induction, there exists a strand e

(n E-lN) of the vn (n E-lN) is non meagre for each n E-lN.

web fJl such that T- 1 e

vn By prop. 1.1.3, since the T- 1 e

T

-1

vn

are absolutely convex, the

are neighborhoods of zero in E : there exist k n vn -1 , Vn f IN. such that uk C T e vn n

e

T

~

Step b. Let >..n(n E-lN) be strictly associated to the strand e (nE-lN). We may assume, by considering a subsequence, that vn An =I= 0 for all n. Using prop. IV.1.8, we can choose An (n E-lN) ~

E An yn is fast convergent for all yn E- e • Finally, vn n=1 changing kn (n E-lN), we may suppose that

such

uk

-1

c

An T

c

T-1 e

c

-1 An T e

n

e

'

vn

vn f lN.

Then uk

vn-1

n

vn > 1 •

'

Indeed, uk

n

Given xn 0 f

Uk

and Yn f T- 1 e

64

, n0

vn

c

vn

A T- 1 e + uk n vn n+1

'

Vn •

there exist then sequences xn E- Uk (n>n 0 n

(n~n ) such that o

)

Summing up these relations for n

~

0

n

~

N, we get

N I:

n=n

0

hence

On the other hand, by definition of strictly associated N I:

sequences,

A n Tyn is fast convergent in F to some point

n=n

z

~

e

vn -1 0

.

thus (xn 'z)

0

Since G(T) is fast sequentially ~

G ( T), hence

X

0

n

~

T-1

0

e

vn -1

C

1 OS ed,

and uk

0

we have

< T-1 n

e

0

v n -1 0

SteD c. By prop. IV.1.7, for every neighborhood V of zero in F, A 1e M

~1

C V for n large enough. Hence T(An+ 1 Uk

n

) C V and T is

continuous. Comment. A first extension of prop. IV.5.1 follows from prop. II.3.12 : we can replace the

~-space

E by an ultrabornological

space. The three other natural extensions consist in 1) considering linear relations instead of linear maps, 2) dealing with more general webs, 3) as in prop. I.2.2, replacing E by a Baire space (the repeated use of prop. I.1.2 prevents from taking E barrelled). The next result can be regarded as the general closed graph theorem for webbed spaces. It is the main result of this section. PROPOSITION IV.5.2. ~ E ~ F such

~

E,F be l.c.s. and R a linear relation

that ~(R)

=

E. Assume that E is ultrabornolo-

sical, F webbed and G(R) fast sequentially closed in E x F. Then R is continuous.

65

In particular, i f R is a linear map of E tinuous and if RE is an

~

~ v(R)

1

~

is a linear map of F onto E, R

~-space,

F, it is con-1

is open,

is still true if we assume

~result

is non meagre in E and Yields V(R) = E,

By prop. II.3.12, we may assume that E is an ~-space (and even a Banach space). Let Uk(kE-JN) be a base of neighborhoods of zero in E and ~

!e

v

As in step a of prop.

: v E- I} be a completing web in F, IV.5.1, there exists a strand -1

(n E-lN) of ~ such that R e is non meagre for each n E- JN. vn vn Observe that this requires only R- 1 (F) = V(R) not to be meagre

e

to start the induction process, rather than V(R) Each R

-1

=

E.

e

has thus an interior point (but it is no lonvn ger a neighborhood of zero). Let An(nfJN) be associated to the strand e a subsequence, we may assume that An Choose xn and kn f A

R

n

-1

e

~

such that

vn

We may assume th a t xn ~~ ~'n R-1 e 1

( xn + 2u k

n

)

x~ +

t i on,

vn

: indeed,

; for every x'n in this intersec-

n An 1

2

+

(n fJN). Taking vn 0 for each n.

C A

Uk

n

n

R

-1

e

vn

Let V be an absolutely convex neighborhood of zero in F. By prop. IV, 1 • 7, co

I

E

i=n

( *)

Ai yi :

for n large neough, say n

~

n

0

•

Let us prove that 2R- 1 V ) Uk

• Since n

0

'In

66

f lN

,

we determine by induction sequences yn E- Uk

given yn

n

0

(n~n ) such that (n>n ) and xln E- A. n R- 1 e 0 vn o X 1

-

n

X

+ y

n

n+ 1

t

';In ~ n

o

Thus N L: n=n

in E.

(X I -X

n

n

)

-o

y

0

n

0

On the other hand,

that

there exist z

1 Rz n and xn1 Rz n" Moreover

N L:

n=n

n

,

z

1

n

e

E- A

n

v

such n

(z 1 -z ) is fast convern

n

0

(*).

gent in F and its limit z belongs to 2V, by

Thus yn

Rz 0

(because G(R) is fast sequentially closed) and y

n

E- 2R-1v. 0

PROPOSIT:ON IV.5.). Prop.IV.5.2 is still true when E is an inductive limit of Baire spaces, i£ G(R) is closed instead of fast seguentially closed. It is also true with no assumption on E if ~(R) is non meagre in E and it yields ~(R)

= E.

The latter condition is an improvment of "~(R)

E" but not

of the assumption on E since, if ~(R) is non meagre in E, E is non meagre in itself, hence a Baire space. The proof has to be modified as in prop. I.2.2. Repeating the above arguments, we obtain a strand e

(n E-lN) vn of fll and a sequence of neighborhoods Un(n E-lN) of zero in E sud that

un

-

X

n

• However Un(n E-lN) is no longer a base of vn neighborhoods of zero in E. e

there exist

Following the preceding proof, given yn 0

x 1 E- An R- 1 e (n~n o ) such that n v n

67

Yn = x~- xn + Yn+1' vn ~no • Let U be an arbitrary neighborhood of zero in E. We have N E

n=n 0 Since

we Hence N

E (x~-xn) - yN+ 1 + xN+ 1 f n=n 0

u.

On the other hand, if yN+ 1 R wN+ 1 , with wN+ 1

~

XN+ 1

N E

n=n 0 is still convergent in F and its limit z belongs to 2V. Thus, since G(R) is closed, we have again (yn ,z) f G(R) and Yn

f

2R- 1 V. Hence the proof.

0

0

Let us come back to prop. IV.5.2. We have seen in prop. IV.5.3 that the assumption on E can be replaced by an assumtion on ~(R), but this is not a real progress on E since anyway it must be a Baire space. Can we now relax the assumption on F? A first obvious answer is that it is enough to assume that R(E) is webbed, instead of F, since it makes no difference to replace F by R(E) • When E is an U-space and F is webbed, E x F is webbed (prop. IV.4.8) hence, if G(R) is fast sequentially closed in E x F, it is also webbed (prop.IV.4.1). So it would be an improvment to take G(R) webbed instead of assuming that G(R) fast sequentially closed and F or R(E) webbed. The eventual progress is however not significant with respect to F since, if G(R) is webbed, R(E) is webbed because it i the image of G(R) by the canonical projection of E x F onto F

68

(prop. IV.4.2). It is however an improvment on G(R), since it can be webbed without being closed nor fast sequentially closed. PROPOSITION IV.5.4. ~ E ~ F.

1L

~(R)

~

E,F be l.c.s. and R a linear relation

is non meagre in E ~ G(R) webbed~ R ~

continuous. of E into G(R) by xR'(x,y) ~ ) xRy. It is still linear and ~(R) = ~(R•). Moreover, R•- 1 is simply Define a new relation R'

the canonical projection and G(R

1 )

n1

of G(R) in E. Thus it is continuous

is closed. By prop.IV.5.3, R' is thus continuous.

This implies the continuity of R. Indeed, R

~

n 2 o R 1 , where

n 2 is the projection of E x F onto F.

6. LOCALIZATION PROPERTIES A simple look at the proof of prop. IV.5.1 shows that, when E is an v-space, F a strictly webbed space and T a linear map of E into F with a fast sequentially closed graph, we get inclusions of the type "TU C e" where U is a neighborhood of zero in E and e an element of the web of F. This has been shown to imply the continuity of To But, since the elements of the web may be much smaller than neighborhoods of zero, it is worth to see what extra information this brings on T. PROPOSITION IV.6.1. tion of E

~

~

E,F be l.c.s. and R be a linear rela~(R)

F such that

= E. Assume that E is an V-spa-

that F has a strict web ~ = {e : v f I} and that G(R) ~ v fast sequentially closed in E x F.

~'

Then, i f -----

v

>

R

-1

e

~

is a neighborhood of zero in E.J there exists -1

-

~ such that R e is also a neighborhood of zero in E. v In particularJ there exists a strand e (kfm) such that all vk -1 R e are neighborhoods of zero and we can choose for the first vk -1 element any v such that R ev is a neighborhood of zero.

69

Assume that R

-1

e

~

is a neighborhood of zero in E. It is thus

non meagre. Since R -1 e that R

-1

u R-1 e 'II

~

,

v>~

there exists v 1

is non meagre too. v1 By induction, we can thus find a strand e

R

-1

>

such

~

e

vn

(n E-lN) such that

e

+¢ for each n. Choose then a strictly associated sequenvn ce An (n E-lN) and, assumin,g that >..n =I= 0 for each n, a base of

neighborhoods of zero U (n E- JN) in E such that Un n for each n.


.

T.

There are interesting examples of reflexive webs, even in non semi-reflexive spaces. For instance, we have seen that the dual E 1 of a metrizable space or of a countable inductive limit of metrizable spaces is strictly webbed (prop.IV.3.2-3). Moreover, its web

~

is such that every

~-bounded

subset is equicontinuous.

It is thus relatively weakly compact when E 1 is equipped with the Mackey topology (or any coarser Hausdorff topology). case, the topology t~ defined above on (Ei)'

=

In this

E is simply the

natural topology of E (every ~-bounded set is equicontinuous an~ conversely, every equicontinuous set is contained in a bounded Banach disk, hence ~-bounded by lemma IV.6.8). COROLLARY IV.6.14.

~

E be evaluable and complete and F be me-

trizable or a countable inductive limit of metrizable spaces.

l£ T is a linear map of E' into F 1 such that G(T) is closed in E 1 x F', 't

~Tis

the adjoint of a continuous linear map ofF

't

i!!..l.Q.

E•

It implies in particular that T is continuous for a lot of topologies on E 1 and F 1 (for instance of E's into F's' of El, into F and so on) • It is a straightforward consequence of cor.IV.6.13 and of the

b,

above comments on the web ofF'.

75

V Webs in spaces of linear maps

1. SPACES OF LINEAR MAPS DEFINITION V.1.1.

Let E,F be l.c.s. We denote by L(E,F) the spa-

ce of all continuous linear maps of E into F. If family of bounded subsets of E, we denote by

t~

~

is a covering

the topology de-

fined on L(E,F) by the semi-norms rB

,p

where B f

=

(T) ~

sup p(Tx) xfB

and p is a continuous semi-norm on F.

The space L(E,F) equipped with the topology t~ ,t~ , ••• is s b denoted by Ls(E,F), Lb(E,F), •••• Moreover, we denote by p the finest locally convex topology on L(E,F) for which the equicontinuous completing disks of L(E,F) are bounded. It is clear that p is finer than t~ for each

~.

The next example will help to see how to find webs in spaces of linear maps. EXAMPLE V.1.2.

1L

E is metrizable and F ~~-space, Lp(E,F) ~

a strict web. Let Uk(k flN) and Vk(k flN) be bases of neighborhoods of zero in E and F. Define e(k) n

!T f

L(E,F)

The sets ( 1)

e(n1, ••• ,ni) = e n1 determine a

76

web~

( i)

n ••• n e n. l.

of L(E,F).

Given a strand e

vi

= e(

n 1 , •• , , nk. )

(n

k

E-lN, k.

l.

T co)

of fll,

l.

for all i, we have c vk,

11i ""'k ,

thus the sequence T.(iE-lN) is equicontinuous. Since F is coml.

plete, every closed and absolutely convex equicontinuous subset of L(E,F) is completing. The conclusion follows by prop,IV.1,8 and 1 o, The localization properties (IV,§6) suggest that the elements of a web of F could play the same role as the Vk ( k E-lN) in the above example, PROPOSITION V,1,3,

1£

E i s a Banach space or a countable induc-

tive limit of Banach spaces and if F has a strict web,

then

Lp(E,F) has a strict web. Assume first

that E is a Banach space, with closed unit ball

B, and that F has a strict web fll, By prop.IV.6,1, for each T E- L(E,F), there exists v E- I and m E- lN such that TB C m e there exist v'

>

and, if this is true for v and m, v v and m' E- lN such that TB C m' e 1 • v

According to prop,IV.1.6, we can change the web fJl in such a way that we no longer need the constants m,m•. Define then

&v

{T E- L(E,F) : TB C e } (v E- I), v

We have just seen that L(E,F) = U & v

v

and &

Jl

&

v

hence fll' {& v E- I} is an absolutely convex web of L(E,F), v It is a strict web. Indeed, take a strand & (k E-lN) and a vk sequence xk(kE-JN) strictly associated to the strand e (kE-lN). vk N I f Ilk E- [o,xk] and Tk E- & { E Ilk Tk : N E- lN} is equicon\lk ' k=1 tinuous. Indeed, for every neighborhood V of zero in F, we have

77

N E

k=k 0 for k 0 large enough (prop.IV.1.7). Thus N

( E ~k Tk)B C V, VN ~ k 0 k=k 0 There exists C

>

0 such that

i ~k

E k=1

Tk B C CV, Vi ( k 0

hence N I:

~k Tk B C (1+C)V,

VN E- lN •

k=1 Moreover, if x E- B, E Tkx E- e

vk

~k

Tk x is convergent in F, since

• Hence E ~k Tk is convergent in Ls(E,F). Since

E

k=k +1

~k

0

Tk x E-

I:

k=k +1 0

~k

e

vk

C e

vk

,

vx E- B ,

0

we also have

We have thus proved that ~· is a strict web of Ls (E,F). Refering to the proof of prop.IV.1.9, for ek- o,

( *) is even fast convergent and, more precisely, convergent in the Banach space generated by the Banach disk

It is elear from the above arguments that K is moreover equicontinuous in L(E,F). Thus it is a bounded Banach disk of L~(E,F) and (*) is also convergent in L~(E,F) hence ~· is a strict web of L~ ( E, F). Assume now that E is the inductive limit of the Banach spaces Ei(i E-lN).

78

The space L (E,F) is isomorphic to the closed subspace of s

00

n

i=1 T.

l.+

L (E. ,F) made of all sequence T. ~ L(E. ,F) (i flN) such that s l. l. l. 1 restricted to E.l. equals T.1 for each i ~lN. By the permanence properties (prop.IV.4.8 and prop.IV.4.1).

Ls(E,F) has thus a strict web~·. If we refer to the construction of this web, it is easy to see that, for each strand &1

vk

(k E-lN) of ~·,

there exists an associated sequence >..k(k E-lN)

such that 00

00

{E

£kJ..kTk:

k=1

E k=1

l£kl~ 1 }

is still equicontinuous in L(E,F) for all Tk ~ 0: 1 (kE-lN). So we vk

can conclude as above that ~· is still a strict web in Lp(E,F). PROPOSITION V.1.4. lL E is an U-space or a countable inductive limit of U-spaces and F a sequentially complete and webbed space or a countable inductive limit of such spaces Fi'

~

LP(E,F) is webbed. If moreover F or each Fi has a closed and absolutely convex ~' ~ LP(E,F) is strictly webbed.

Assume first that E is an U-space and F a sequentially complete webbed space. Let U. (i ~lN) denote a base of neighborhoods of zero in E and ~ = {e

1

v

: v ~ I} a completing web of F.

If T E- L(E,F), by prop.IV.6.3, t"nere ex1.sts · · v E- I sue h that T- 1 e v l.S non meagre, if T- 1 e

v

is non meagre, there exists

~

>

v

such that T

-1

e

~

is non meagre and i E-lN such that TU.1 C ~(e v ). Define then 0:(. ) (. ) (where each (i 1 ,n 1 ), ••• , 11,n1 , ••• , 1k,nk (ik,nk) should be indexed by a single index inlN) as the set of all T E- L(E,F) such that T that

-1

e(

n1'"""'nk

) is non meagre and

79

The sets

define a web~· in L(E,F).

&(.l.1,n1 ) , ••• , (•l.k,nk )

(k ~lN) in 9i 1 • The corresponding indices vk nk(k ~lN) determine a strand in 9i. Take a decreasing sequence

Fix a strand C

Xk(k ~JN) associated to the strand e( n 1 , ••• ,nk )(k~lN) of 9i. If Tk ~ C

(k ~lN), the sequence Xk Tk (k ~lN) is equicontinuous. vk Indeed, for every neighborhood V of zero in F, by prop.IV.1.7,

for k large enough, say k

~

k , 0

Xk e( n , ••• ,nk ) C V 1

hence, if we choose V closed and absolutely convex, C

Cc;"(e( n , ••• , nk )) C V, lfk ~ k o • 1 0

Since F is sequentially complete, every closed, absolutely convex and equicontinuous subset of L (E,F) is a Banach disk. s

It is also a Banach disk in L~(E,F). Thus, by prop.IV.1.8, 9i 1 is a completing web of L~(E,F). If the elements e of 9i are closed and absolutely convex, v 1 the condition "T- e v is non meagre and TU.l. C C'O(ev)" is equivalent to 11 TU.l. C e v "• Thus

These are closed and absolutely convex subsets of Ls(E,F), thus, by prop.IV.1.10, the web 9i 1 is strict. Assume now that E is an U-space and F a countable inductive limit of sequentially complete and webbed spaces F. (i ~IN). l. By cor.IV.6.5, L(E,F) is the union of the spaces L(E,F.). l. Moreover, every equicontinuous subset of L(E,F.) is equicontil. nuous in L(E,F). Thus the topology ~ of L(E,F) is coarser in L(E,Fi) than the topology ~ of L(E,Fi). The conclusion follows then from prop.IVo4o5o Assuming finally that E is a countable inductive limit of U-spaces Ei(i~IN), we conclude as in the last part of the proof of prop.V.1.3.

80

Another proof for L (E,F) is obtained by observing that it s is the projective limit of the spaces L (E.,F) and by applying s 1 prop.IV.4o7• However the topology of L~(E,F) is finer that the topology of the projective limit of the spaces L~(Ei,F). The reason why the result is however true is hidden in a slight ~

improvment of cor.IV.4.3 : let disks of E and e

~

be a collection of Banach

be a web of E such that, for each strand

(k f I), there exists an associated sequence J..k(k fJN) such vk

that I:j.Lk ~ (IJ.kf[O,J..k], xk f e space EB for some B f

~.

vk

Then

) is convergent in the Banach ~

is still a completing web when

E is equipped with the topology of the inductive limit of the spaces EB(B f

~).

In the present case,

~

is the family of equi-

continuous Banach disks and the corresponding topology

~

could

be finer than the ultrabornological topology associated to L8 (E,F). The next properties are closely related to the above ones since the dual

E~

of a metrizable space is in a suitable sense

a countable inductive limit [41,91]. PROPOSITION V.1.5.

~

E be metrizable and F admit a strict web

(reap. be sequentially complete and webbed). Then Lb(E~,F) ~ strictly webbed (resp. webbed). Let E~ the inductive limit of the spaces E'u?' where Ui(i flN) 1

is a base of neighborhoods of zero in E. By prop.V.1.3 and 4, L~(E~,F) has a

strict (reap. completing) web when F is strictly webbed (reap. sequentially complete and webbed). The space L(E~,F) is a vector subspace of L(E~,F). Indeed, since

U~

is bounded in E 1 c in F, hence T f L(E~,F). 1

If~

,

for all T f

L(E',F), c

TU~ 1

is bounded

= {e v (v f I)} is a strict (reap. completing) web of F,

the web~· of L(E~,F) consists of finite intersections of sets & (i, v)

= {T

f L ( E, F) : TU

i

C

e) ( i

f :N, v f

I) ,

Provided the constants are ruled out by means of prop.IV.1.6, 81

or of sets {T E- L(E,F) : (T- 1 e ) n Euo is non meagre in v i and TU~ C G;(e )} l. v

Eij~ l.

•

If E' (k E-:IN) is a strand of .7! 1 and A.k(k E-lN) an associated vk sequence, for each i E-lN, there exists a strand .7t such

that

T U~l. C G;(e v (")' VT E- C'vk l. k

for k large enough. Thus, given a neighborhood V of zero in F and £ > o, for all ILk E- [O,A.k] and Tk E- C' vk ' N ( I: !LkTk)Ui_ c e:V, VN ~ k 0 ' k=k 0 for k 0 large enough (prop.IV.1.7). Therefore, if q is the gauge semi-norm of V, N

q [( E !LkTk)x 1 ] k=k 0 is uniformly convergent on cr(E 1 ,E), the limit q[(

""I:

U~. l.

Since it is continuous on

!LkTk)x 1 ]

U~ l.

for

is still continuous on each

k=k 0 Ui_ for

0

(E 1 ,E). It follows then by the Banach-Dieudonne theorem

(prop.I.2.6) that q[( "" I: !LkTk)x 1 ] "" k=k 0 I: 11kTk E- L(E 1 ,F). k=k c

is continuous onE' , so that c

0

Thus L(E~,F) is closed in L(E~,F) for the sequences considered when checking that .7! 1 is a completing web. Hence the web

!C'v

n L(E',F) : C' c v

f

.7!'}

is still completing in L(E' ,F) equipped c

with the topology induced by L~(E~,F). The topology of Ls(E&,F) is coarser than the topology of L~(E&,F) and coincides on L(E~,F) with the topology of Ls(E~,F). Thus Ls(E~,F) is webbed. Moreover, the topology of Lb(E~,F) is coarser than the ultrabornological topology associated to Ls(E~,F),

82

thus Lb(E~,F) is also webbed (cor.IV.4.)).

PROPOSITION V.1.6.

~

E be a strict inductive limit of metri-

zable spaces Ei ~ F a strictly webbed (resp. sequentially complete and webbed) space, such that, for every sequence of continuous semi-norms q.l. (i E-lN), there exists a continuous semi-

>

..!.UU:.!!l q and a sequence Xi

0 (i E-lN) such that

qi(x) ~Xi q(x), ~if lN, ~ Lb(E~,F)

~x E- F

•

is strictly webbed (resp. webbed).

Define ~ 1. of L(E! ,F) into L(E',F) by l. 'c c (~iT)x 1

=

T(x• IE.), ~x' f E•

•

l.

We shall prove that 00

L(E',F) = C

U

i=1

"t.L(E! l.

J.,C

,F) •

its restriction to E.l. is For every bounded subset B of E'c clearly bounded in E! • Thus each ~ 1. is continuous of J.,c Lb ( E ! , F ) in t o Lb(E~,F). The conclusion follows then from prop. J.,c

IV.4.5 and 2. LetT belong to L(E',F). For some i f lN, TE~l. = o. Indeed, c otherwise, there exist for each i f lN, x.l. f E!l. and a continuous semi-norm qi ofF such that qi(Txi)

I

0. There exists then a

continuous semi-norm q of F such that q(Tx.) l.

+0

and it is not a restriction to assume that q(Txi) i

(multiply x. by a suitable constant). However, l.

for each i E-lN

=

1 for each

the sequence

xi (i flN) is equicontinuous on E. It is moreover convergent to zero in E', thus, because of the equicontinuity, in E'c• Hence a s contradiction. 0. We construct T. f L(E! ,F) such that Assume that TE~ l.

l.

~iTi

J.,C

=

T. Since the inductive limit is strict, the topology induced in

Ei byE is equal to the topology of Ei. Thus Ei. = {xiE.: x E-E'}. l.

Given x f Tix

=

Y-y 1 f

El, take y f

E 1 such that YIE.

=

x and define

Ty. The definition is coherent, sine~ YIE.

Y'IE.

l.

l.

implies

Ei hence Ty - Ty' = 0.

83

It is clear that T.~ is linear. To prove that T.~ is continuous, by the Banach-Dieudonne theorem, it is enough to check that for each continuous semi-norm q on F, q(T.x) is continuous ~ for a(E!,E.) in each equicontinuous subset A of E!. If it is not ~

true,

~

there exist E:

>

~

0

and a generalized sequence xa( a E- €7) E- A

such that q(T.x ) ~ E: and x ( ~ e1)- 0 for cr(E! ,E.). By the Hahn~ a aal 11n x~K

x~Kn

'

hence the result. If moreover F is complete, c;(K) is still compact. PROPOSITION V.1.9. The class Q is stable for a) subs paces, b) quotients, c) countable inductive limits and direct sums, d) finite products. a) If L is a vector subspace of E, a neighborhood of zero in L is the restrict1on to L of a neighborhood of zero in E, hence the result. b) If E/L is a quotient space and j

the canonical quotient

map, every neighborhood of zero in E/L is the image by j

of a

neighborhood of zero in E. Hence the conclusion.

c) Let E be the inductive limit of the spaces Ei (i ~lN) and Uk(k ~m) a sequence of neighborhoods of zero in E. There exist neighborhoods

U~~)

of zero_in Ei such that Uk )

U~i)

for each

There exist then A.~ 1 ) > 0 and a neighborhood Vi of zero

i ~lN. in each Ei such that

~ k=1

A.(i) k

u(i) ' k

85

Put 1/sup A(i)

lli

k

lc.!;;i

and

00

v = Co(

u

llivi).

i=1 Then uk

)

u(i) k

and

)

1

0TI k

v.l.

)

lli vi, Vi

:1!:

k

'

for £k small enough. Thus 00

uk) inf (1,£k) Co( u lliv 1 ), Ilk~ E i=1

,

00

where Co( U ll·V.) is a neighborhood of zero in E. i=1 l. l. d) is straightforward. The following simple argument will provide new examples of webbed spaces of linear maps. Let J be the map T -

T*. For sui-

table topologies on the various spaces involved, J may happen to be continuous and surjective of L(E,F) into L(F 1 ,E 1 reover L(E,F) is webbed, then L(F ,E 1

).

If mo-

is webbed too by prop.

1 )

IVe4o2o

In the sequel of this paragraph,

~.~'

denote any covering

family of bounded sets, which is stable under continuous linear maps, i.e. T~(E) C ~(F) for all T ~ L(E,F). PROPOSITION V.1.10. The map J is defined of L(E,F) ~ L(F~,E~) for all ~.

Indeed, if T ~ L(E,F), for every B ~~(E), TB belongs to the corresponding family ~(F). Thus sup l<x,T*y'>l = sup Il y~TB

F~

into

E~.

PROPOSITION V.1.11. The map J

is continuous of L~(E,F) into

LJ.3(F JJ' ,E~,) for all 9.3 .!!.!ll! 9.3 Indeed, to B ~ 9.3(F~ 1 ) and B• 1 •

~ 9.3'(E) corresponds

the neigh-

borhood of zero IT' ~ L(F.JJ, ,E.JJ,)

w=

sup

sup

x~B'

in LJ.3(F~ 1 ,E~ 1 -1

J

W

=

).

l<x,T 1 y'>l ~ 1}

y 1 ~B

Its inverse image by J,

IT~ L(E,F)

: sup xE-B'

sup

jj ~ 1}

y'~B

is clearly a barrel in L (E,F). Thus it absorbs every Banach s disk of Ls(E,F) and it is a neighborhood of zero in L~(E,F). PROPOSITION V.1.12.

I£ E is evaluable, the map

J

is surjective

££ L(E,F) ~ L(F~,E~) ~ t9.3 is coarser than t9.3

ill F•. 't"

let T 1 belong to L(F~,E~). For every y'

Indeed,

~ F•,

T'y'

is a linear form onE and, for each x E- E, y ' - <x,T'y'> is a linear form on F 1 • ce t9.3

~

(x,T y 1

1

t9.3 , >

't"

=

It is moreover continuous on F.JJ, hence, sin-

there exists a unique element Tx (Tx,y

1 )

~

F such that

for ally E- F'. We define this way a linear

map T of E into F such that T*

=

T•. The map Tis continuous.

Indeed, Tis continuous of Eo(E,E') into Fo(F,F•)• Thus it maps bounded subsets of E into bounded subsets of F (by the Mackey boundedness theorem). If V is a closed and absolutely convex neighborhood of zero in F, T- 1 v is thus absolutely convex, closed and bornivorous. Since E is evaluable, it is thus a neighborhood of zero. Combining the above results, we get the following properties. PROPOSITION V.1.13.

~

E be a Banach space or a countable in-

ductive limit of Banach spaces and F a strictly webbed l.c.s. strictly webbed if t9.3 1 is coarser then t9.3 't"

illF'• By prop.V.1.3, L~(E,F) is strictly webbed, By prop.v.1.11, the map J is continuous of t 13 (E,F). into LJ.3(F9J 1 ,E~, ). By prop. V.1. 12,since E is evaluable and t9.3 1 is coarser than t9.3 in F 1

,

't"

87

J

is moreover surjective. Conclude by prop.IV.4.2.

PROPOSITION V.1.14. Under the assumptions of prop.V.1.4, L~(F~ 1 ,E~ 1 )

is webbed or strictly webbed provided t~ 1 is coarser

ill F .. Same argnment.

.ih.!m t ~

I •

A rough application of the same ideas to prop.V.1.5 does not bring any new result. Indeed, if E is metrizable and E' evaluac

ble, it is equal to Eb and as we have seen in the proof of prop. III.).12,Eb is a countable inductive limit of Banach spaces. A more careful approach can however lead to some interesting results. PROPOSITION V.1.15.

E be metrizable and F admit a strict web

~

(resp. be sequentially complete and webbed). ~ Lr(F~a'E) ~ strictly webbed (resp. webbed), where r is the class of eguicontinuous subsets of F 1 T

1

•

The map J is sur,iective of L(E 1 ,F) into L(F 1 ,E). Indeed, let c ca belong to L(F 1 ,E). For each x 1 f El, ca y

I

-

(T

I

y t

t X I)

is an element of (F'ca ) 1 , hence of F. Define T by l

1 is a continuous semi-norm of E c•

Moreover, Lr (F'ca ,E) sup y 1 fV 0

T' is continuous of Lb(E~,F) into if U and V are neighborhoods of zero in E and F, the map J

sup fU 0

X 1

: T

I<JTy 1 ,x'>l

sup sup fU 0 y 1 fV 0

Il

X 1

is a continuous semi-norm of Lb(E~ 1 F). The conclusion follows from prop.V.1.5.

88

2. TENSOR PRODUCTS We do not have any direct information about the permanence properties of webbed spaces with respect to tensor products. However tensor products can be looked at as subspaces of spaces of linear maps. When they are closed subspaces, it is thus possible to use §1 and prop.4.1. For the general theory of tensor products of l.c.s., we refer to [95]. PROPOSITION V.2.1. The tensor product E

~

E

F of two l•c.s. is

a subspace of Lr(E~a'F), equipped with the induced topology, where r is the family of eguicontinuous subsets of E•. Moreover Lr(E'ca ,F) is a closed subspace of Lr(E ~1 ,F). It is well known from algebra that the map x ® y where

Tx ® y'

Tx ® y x' = <x,xl)y extends to a linear injection of E ® F into L(E 1 ,F), provided E 1 is equipped with a topology t such that (E 1

~

By definition, the topology

E

of E

~

) 1

=E.

F is the topology of

uniform convergence on equicontinuous sets of Et and F 1

,

so it

is the topology induced by Lr(E~,F). LetT belong to L(E 1 ,F). We have ~

T E- L ( E 1

ca

,

F)



T * E- L ( F

1 ca , E ) •

Indeed, if T E- L(E~a'F), then T* E- L(F~,(E~a)~). But (E~a) 1 = E and, for 9J = 9J , the topology of (E 1 )~ is finer than the inica ca ~ tial topology on E. Conversely, if T* E- L(F~a'E), (T*)* coincides with T hence, by the same argument, T E- L(E~a'F). Let now U be a closed and absolutely convex neighborhood of zero in E. We have T*-1

u

{y I E- Fl {y I E- Fl

I I

I a

and x ' -

u A.l. i=1

A

and 00

00

A

)

~

i=1

A.i Ai

=>

A

)

~

i=1

A.i A.l.

93

it follows easily that

{e v :

v f

I}

is a new web which is still

completing or strict. For each~ f

L00 (e,E), C~ ~(e) is a bounded Banach disk of E. or 3 to the identical map of the corres-

Applying prop.IV.6.1

ponding Banach space into E, we get, -

there exists v f

- i f Co ~(e) C ev' and, when -

when~

is strict,

I such that C~ ~(e) C ev'

>

there exists v 1

v such that Co ~(e) C ev''

is only completing,

~

there exists v f

I such that e

n Ec; ~ (e) is non meagre in

v

Ec~ ~(e) and c; ~(e) C Co ev -

if the preceding statement is true for v, it is true for some

>

\1 I

V o

Assume that ~ is strict. Then ~·

= {&

v

I} , with

v f

Co ~(e) C e } v

is an absolutely convex web of L~(e,E). Given a strand &

\lk

(k flN),

let A.k(k ElN) be strictly associated

to the corresponding strand of to & X

f

(k ElN). Indeed, given Ilk f

vk e,

00

I t is also strictly associated

~.

[o,>..k] and

~k

f

&

vk '

for each

Ilk ~k(x) is convergent in E and

~

k=1 ~

!

~

k=k +1

Ilk ~k(x)

x

f

e}

C

0

00

By prop.IV.1.7,

I:

k=1 limit of

N I:

k=1

Ilk

~k

Ilk

~k

is thus an element of L00 (e,E) and the

for the topology of L00 (e,E). Thus ~·

completing web. We have moreover

We must improve this and show that

94

is a

co

co

( I: k=k 0 +1

c

Ilk IPk)(e)

e

,

vk

lfk 0 flN

0

We have co

co

11k c"'o q~k(e)

I:

c

Given C

>

k=N

e

VN-1

,

>

liN

1 •

neighborhood U of zero in E, there exist

~arbitrary

0 and N

c

Ilk e vk

I:

k=N

k +1 such that e C CU. Hence o VN

~

N E 11k Co IPk(e) + cu k=k +1 0

co

E 11k c~ IPk(e) k=k +1 0

c

c

c•u

for C 1 large enough and co

,...

Ilk Co IPk(e)

I:

k=k +1 0

is bounded in E for each k 0 f

~.

It is thus contained in a

Banach disk B. co

co

Given x f Co ( I: Ilk IPk)e, we can write it x = I: Am xm' k=k 0 +1 m=1 co

with x

co

I:

m

k=k +1

Ilk xk

0

,m

(xk

,m

f

IPk(e),

I:

m=1

IAmi

~ 1 ). The

series co

co

I:

I:

m=1 k=k +1

A

m

11

X

k

k,m

0

is absolutely convergent in EB. Thus co X

=

co

co

I: Ilk Co IPk(e) • Ilk ( I: Am xk ,m ) f k=k +1 k=k 0 +1 m=1 0 I:

Therefrom, 00

00

Co ( I: Ilk IPk)(e) k=k 0 +1

c

I:

k=k +1

Ilk Co IPk(e)

0

00

c

I:

k=k 0 +1

Ilk e vk

c

e

,

vk

Ilk

0

flN

.

0

95

When

~

is only a completing web, the web

~·

is the collection

of {cp E- L00 (e,E)

e v n E"" Co cp(e) is non meagre in Ec""o m(e) T an d cp ( e ) C C o ( e ) } • v

The proof is similar to the first part of the above one but

~·

is no longer strict. DEFINITION V.3.5. Let e (m E-lN) be an increasing sequence of subm sets of e. We denote by Loo (e m(mE-lN),E) the set of all maps cp of e into E such that cp(e ) is bounded in E for all m E-lli, equipped m with the topology of uniform convergence on the subsets e (m E-lN~ m COROLLARY V.3.6. !£ E is fast complete and webbed (resp. strictly webbed), .l!:!2 L00 (em(m E-JN)'E) is webbed (resp. strictly web~).

Indeed, L00 (em(mE-lli)'E) is the projective limit of the spaces L (e ,E) (mE-lN). Apply then prop.Vo3o4 and IV.4.7. "" m REMARK V.3.7. In prop.V.3.4 and cor.V.3.6, the assumption that E is fast complete is only used to assert that cp(e) is contained in a Banach disk. So both would hold true without the fast completeness of E, for the subspaces L:,(e,E) and L:,(em(m E-JN)'E) of all maps cp of e into E such that cp(e) (or cp(e )) is contaim

ned in a bounded Banach disk of E (for each m). There are many interesting examples of subspaces of L00 (e,E) or L00 (em(m E-IN)'E). DEFINITION V.3.e. We denote by l(E) the set of all sequences x (m E-lN) of E, equipped with the topology of pointwise converm

gence, defined by the semi-norms n(x ( E-lN)) m m

=

sup p(x.) , i~N

where p is a continuous

96

~

~emi-norm

of E and N E- IN.

The space l~(E) is the space of all bounded sequences, equipped with the topology defined by the semi-norms n(x (

LJN)) = sup p(x ) , mE-lN m

mm~

where p is a continuous semi-norm of E. The space c 0 (E) is the subspace of l~(E) of all sequences convergent to zero. PROPOSITION V.3.9. ~ E be webbed (resp. strictly webbed). ~ l(E)

is webbed (resp. strictly webbed).

l!

E is moreover

fast complete, l~(E) ~ c 0 (E) are webbed (resp. strictly webbed). The space l(E) is l 00 (em(mE-lN)'E), where em= For each 4l E- l,)em(m

e-m) ,E),

{1, ••• ,m).

4l(em) is finite, hence contained

in a bounded Banach disk. The conclusion follows from cor. V.3.6 and remark V.3.7. The space 1 00 (E) is 1 (lN,E) and c 0 (E) is a closed subspace ~

of 1 00 (E). Regarding spaces of continuous or differentiable functions, we will restrict our investigations to two striking examples. Many others can be handled by the same arguments. DEFINITION V.3.10. Let E be a compact space and E a l.c.s. We denote by C(K,E) the space of all continuous maps of E into E, equipped with the topology of uniform convergence on K. Let Q be an open subset of mn. We denote by 8(Q,E) the space of all infinitely differentiable maps of Q into E, equipped with the topology defined by the semi-norms sup sup p(D~ 4l) xE-K jo:j.e;m where E is compact in Q, p a continuous semi-norm of E and Do: X a derivative of~ of order

jo:j.

97

!p

PROPOSITION V.}.11. lL E is fast complete and strictly webbed (resp. webbed) ~ C(K,E) ~ &(a,E) are strictly webbed (resp. webbed). The space C(K,E) is a closed subspace of Lco (K,E). If Km(m E-lN) is a increasing sequence of compact subsets of Q such that K C K0 1 for each m E-m, &(a,E) is isomorphic to a closed m m+ subspace of the product of countably many copies of L00 (Km(m E-li:{)'E), through the embedding which, to each II' E- &(a,E), associates the sequence of all its derivatives (arranged in a given order). Various spaces of linear maps can also be recovered by similar arguments. PROPOSITION V.}.12. 1Al E be a normed space or a countable inductive limit of normed spaces Ei (i E-lN) ~ F be fast complete and webbed (reap. strictly webbed). ~ L~(E,F) is webbed (reap. strictly webbed).

lL E or the E.'s are Banach spaces, F does not need to be l. fast complete. It is an improvment of prop.V.1.}. Let B (reap. Bi) be the closed unit ball of E (reap. Ei). When E is normed, the map J : T- TIB is an isomorphism of Lb(E,F) onto a closed subspace of L~(B,F). If E is the inductive limit of the normed spaces E.(iE-IN), J aT -TI 00 is l. U Bi i=1 an isomorphism of L~(E,F) onto a closed subspace of L00 (~,F), where ~ = {Bi : i E-lN}. Indeed, J is clearly injective. Moreover, J L(E,F) is the set of all II' E- L00 (~,F) such that ~p(

N I:

i=1

ci xi)

=

N I:

i-1

ci ~p(xi)

00

for all N E- 1N' xi E-

u

j=1

Bj and ci such that

is trivially closed in L00 (~,F).

98

N I:

i=1

Ic 1 I

~ 1.

It

Thus, by prop.V.3.4 and cor.V.3.6, Lb(E,F) (or L~(E,F)) is webbed or strictly webbed. If

~·

is the web defined in it

according to prop.V.3.4 or cor.v.3.6, for each strand 8 and each associated sequence A.k(k E-lN), the series

vk

(k E-lN)

N ~

Ilk Tk k=1 (Ilk E- [O,A.k], Tk E- &vk) (n E-lN) are equicontinuous. Arguing- as

in the proof of prop.V.1.3, we obtain

that~·

is a completing

or strict web of L~(E,F). If E is a Banach space or an inductive limit of Banach spaces, use remark V.3.7. The improvment of prop.V.1.3 could be derived by a simple modification of its proof : the main step is that, since B is a bounded Banach disk, prop.IV.6.1 can be used to claim that TB is absorbed by elements of the web of F. If E is for instance only normed but F fast complete, C~o(TB) is again a Banach disk and the proof can be continued with Co(TB) instead of TB. The same argument also provides another proof of prop.V.1.5: using the notation of prop.V.1.5 and its proof, Lb(E~,F) is isomorphic to a closed subspace of L00 (u;(m E-lN)'F). 4• LOCALIZATION PROPERTIES FOR SUBSETS OF L(E,F) Let E be an

~-space

~-spaces Fi(iE-lN).

and F the inductive limit of a sequence of It is well known (cf.cor.IV.6.5) that, if T

is a continuous linear map of E into F, then TE C Fi for some i

E- lN. If we replace T by a subset ~ of L(E,F), under which

conditions is it true that, for a fixed i

T E-

E- lN, TE C Fi for all

~?

The question

~as

been asked by Hirschfeld and solved by

Kothe in [65]. Further investigations are due to Floret [39]. We have seen in chapter IV, §6 that the above property, for a single map, is widely generalized by the localization property (prop.IV.6, 1-2-3). So it is natural to ask a similar question for webbed spaces. The description of the webs of spaces of linear maps provides an easy answer.

99

PROPOSITION V.4.1.

=

~ ~

E be a Banach space and F have a strict

~

{ev : v ~I}.

l£

B is the closed unit ball of E ~ $

a bounded Banach disk of L (E,F), s - there exist v

- iL

~

I

~

C

> 0 such that

$B C Ce , there exist v'

) v

v

~

~B

>

C1

C Cev 0 such that

$B C C 1 e ,. v

- in particular, there exist a strand e l l Ck

>

0 ( k ~ lN) such that

Let us first modify the

vk

(k ~

m)

and a sequen-

for each k ~ IN. vk according to prop.IV.1.6 in

B C Ck e

web~

order to get rid of the constants. By prop.V.1.3, ~

= {8 v

1

8

v

: =

v ~ I} , where {T ~ L(E,F) : TB C e } v

is a strict web of Ls (E,F). The conclusion follows from prop.IV.6.1, applied to the identical map of E PROPOSITION V.4.2.

~

into L (E,F). s

~

E be normed and F fast complete and

equipped with a strict web

~

=

{e

v

: v ~ I}.

l£

B is the unit

ball of E ~$a bounded Banach disk of L (E,F), there exist s a strand ev (k E- lN) and Ck > 0 (k E- IN) such that k

for each k ~IN. vk Same argument, using this time the web of L (E,F) provided s by prop.Vo3o12.

$B C Ck e

COROLLARY Vo4•3• In both prop.V.4.1

~

2, assume that F

~

the inductive limit of a sequence of strictly webbed spaces Fi(i E-IN). Then there exists k ~lN such that $E C Fk and that $

is equicontinuous in L(E,Fk).

i

Equip F with the web of prop.IV.4.5 1 e{i} = Fi for all ~lN. It is then immediate that $B (hence $E) C Fk for some k.

The equicontinuity follows from the fact that $

is then bounded

in Lb(E,F), hence equicontinuous (since E is normed).

100

PROPOSITION V.4.4.

~

E be a Baire space and F be fast com-

plete and admit a strict web~= {e : v ~I}. Denote by U A v base of neighborhoods of zero in E.

l!

~

(k ~ m) vk for each k ~lN.

is bounded in L (E,F), there exist a strand e s

and a sequence Uk(k ~ lN) .iJl U such that ~Uk C ev k

We assume again

that~

is modified according to prop.IV.1.6.

This time, there is no web in L(E,F) that could be used in the proof. We proceed as for the closed graph theorem. For each x ~ E, ~x is bounded in F, hence Co(~x) is a bounded Banach disk of F. Applying prop.IV.6.1 map of F ~·

~

Co

(

~x

to the identical

) into F, it is easy to see that the collection

of g

(v ~ I) C0(:1.3x) ( e } = {x ~ E v v is an absolutely convex web in E. Thus there exists a strand g is non meagre for each k ~ m. (k ~ IN) of fl?. 1 such that g vk vk Let Ak(k ~m) be strictly associated to the strand e

vk

(k ~IN). There exist Uk f

U(k E-IN) such that

A fortiori,

By the proof of the closed graph theorem (prop.IV.5.1) adapted to the case when E is a Baire space (prop.IV.5.3), it follows that Uk C T

-1

e

vk-1

, Ilk

~

m,

'IT E-

~

henne that , Ilk E-IN. Hence the result.

1 01

PROPOSITION Vo4o5o

~

E be metrizable and F be fast complete

and admit a strict web~~ {ev : v f

I}.

1L

~is

eguiconti-

nuous in L(E,F), - there exists v f

- i!

~E

C >e

v

ev< ,

there exists v' > v such that {me

v

1

:

~E

C )e

v

,< •

m f lN, v' > v} is a strict web of

)e (. Thus, it is enough to prove that there exists v ~~ v

such that

~E

C )e

v



=

0

0

Moreover, X

E- R(T*)

0

)

o, o,

Vy I E- ~(T*) Vy I E- ~(T*)

The last condition is true i f X E- ker T and equivalent to X E- ker T i f ~(T*) is a(F•,F)-dense in F I 0

PROPOSITION VI.1.}. ~ T be a linear map of E into F. It is relatively open if and only if, for every continuous semi-norm p ~ E, the semi-norm

1 0}

q(Tx)

inf

p (X+ X I )

x 1 ~ker

T

is continuous on R(T). Indeed, T is relatively open if and only if every absolutely convex neighborhood U of zero in E is mapped onto a neighborhood of zero in R(T).

If p is the gauge semi-norm of U,

the

gauge semi-norm of TU is q, hence the result. PROPOSITION VI.1.4.

T be a linear map of E into F;

~

a) T is relatively weakly open if and only if R(T*) = (ker T)

0

,

b) i f G(T) is closed in E x F, T is relatively weakly open if and only if R(T*) is o(E•,E)-closed. a) By prop.VI.1.2, we have ker T C R(T*) R(T*) C (ker T)

0 •

Take now y ~ (ker T)

<x,y> if Tx

<x',y'>

=

x'

0 •

thus

0 ,

Define y'

on R(T) by

•

The definition makes sense since Tx Now y'

=

Tz => x - z

ker T

~

) <x,y> = •

is a linear form on R(T). If T is relatively weakly open

or relatively open, y' [x' ~ R(T):

is continuous : indeed,

j<x',y'>l < 1} = T({x~E: j<x,y>j < 1})

is open in R(T). By the Hahn-Banach theorem, y' is the restriction to R(T) of a continuous linear form y" on F. From l)•

~

J

There exist xi(i~p) in ker T such that (xi,xj>=oij(i,j~p). Then, for each x

E,

~

p

q 1 (Tx)~sup(sup i~p ~sup p ~~n


xo,x!>l, sup l<x- l: (x,x!>xo,y!>l) j=1 J J ~ p(i~n j=1 J J ~

l<x,y!>l ~

Since y!~ ~ (ker T)

0

= R(T*), there exist z!~ (pl ~ 1, ~yt ~ vo}

{x

~

E

Tx ~ yoo} = T-1V

and, by prop.VI.1.4.a),

Hence T- 1 v


0 such that

~

m such

n R(T*)•

There-

0 U0 n R(T*) C AT* Vm

and U0 hence U0

n

R(T*)

n

0 = U0 n AT* Vm

R(T*) is closed in Et for a(Et,E).

c a follows from the open mapping theorem (prop.I.3.2). For the converse, observe that T induces an isomorphism between E/ker T and R(T). Since E/ker T is complete, R(T) is also complete, hence closed in F. Condition (f) is, as far as we know, due to Edwards in a slightly different form [35, thm 8.6.13]. The material above is widely inspired by [35, §a.6]. 2. SURJECTIVITY THEOREMS

The results of §1 provide various conditions for the surjectivity of T and T*, by observing that R(T) = F is equivalent to R(T) is closed and dense and that R(T) is dense if and only if T* is injective. First, prop.VI.1.8 characterizes the surjectivity ofT and of T*. For ~-spaces, we can particularize prop.VI.1.10. PROPOSITION VI.2.1. ~ E ~ F ~~-spaces and T a continuous linear map of E ~ F. The following conditions are equivalent: 110

(a) T is sur,jective, (b) T is open,

(c) T is weakly open, (d) T* is a weak isomorphism of F• ~ R(T*), (e) T* is injective and R(T*) is fast sequentially closed in E1

.£..!2£ cr(E•,E).

(f) R(T) is closed nnd T* is injective. Straightforward consequence of prop.VI.1.10. PROPOSITION VI.2.2.

~

E

~

F

~~-spaces

and T a continu-

ous linear map of E into F. The following conditions are equivalent (a) T* is surjective, (b) T* is weakly open, (c) T is a weak isomorphism of E ~ R(T), (d) T is an isomorphism of E ~ R(T), (e) R(T) is closed in F ~ T is injective, (f) R(T*) is closed (or fast sequentially closed) ~ E' for cr(E 1 ,E) ~Tis injective. By prop.VI.1.2, R(T*) is dense in E' if and only if R(T*) s

0

=

ker T = !O}. To be equal toE', it must moreover be closed in E'. The result is then a simple consequence of prop.VI.1.10. s

The next result is inspired by Treves [104,

thm 37,2] quo-

ted in prop.VI.2o4o PROPOSITION VI.2.3. linear map of E

~

~

E,F

~

~-spaces

and T a continuous

F. Assume that, for every neighborhood U ~ F such that

of zero in E, there exists a vector subspace N R(T) + N ~ R(T)

=

F

~

uo n

R(T*) C T*N° (resp.T*

is closed in F (reap. R(T)

=

-1

U° C N°).

F).

It is clear that T*- 1 U° C N° ( ) U0 n R(T*) C T*N° and ker T* C N° Since ker T* 1 11

ker T* C N° (~ N C

RTTJ • F, it implies that R(T) is dense in F.

When moreover R(T) + N

So it is clearly enough to prove the first assertion. We may assume that N is closed. The space F/N is still an ~-space.

If n is the quotient map, n o T is continuous of E

= F/N is closed in F/N. By prop.VI.1.10,

into F/N and R(noT)

R[(noT)*] is thus a(E•,E)-closed in E•. By the canonical identification of (F/N)t and No, (noT)* corresponds to T*INo• Thus T*N° is a(E 1 ,E)-closed in E 1 • Therefrom,

n

U0 n R(T*) = U0

T* N°

is a(Et,E)-closed in E' and, by the Krein-Smulian theorem, R(T*) is a(E•,E)-closed, hence the result, by prop.VI.1.10. PROPOSITION VI.2.4. linear map of E

~

~

E,F

~ ~-spaces

and T a continuous

F. The following conditions are equi-

valent : (a) T is surjective, (b) for every neighborhood U of zero in E, there exists a vector subspace N R(T) + N

~

F such that

= F

T*- 1 U° C N° •

(c) there exists a non increasing sequence Nk(k E-lN) of closed subspaces ofF such that R(T) + Nk

= F for each k

00

n

Nk =

k=1

{o}

and that, for every neighborhood U of zero in E, T ~

a

=>

-1

Nk C ker T + U

k large enough. b and c

TakeN= Nk = {0},

112

e-m,~

b

)

a

It is proved in prop.VI.2.3. c

=>

a

Let Uk(k E-lN) be a base of neighborhoods of zero in E such that 2Uk+ 1 C Uk for each k E- lN. By a suitable relabelling, we may assume that T- 1 Nk C ker T + Uk and R(T) + Nk

=

F, Ilk E-lN •

Given y E- F, for each k E-m, there exists yk E- Nk such that y - yk E- R(T). Hence yk- yk+1 = (y-yk+1) - (y-yk) E- R(T) n Nk C TUk • OCI

Choose xk E- Uk such that Txk = yk+ 1 • The series

E xk is conk=1

vergent to some x E- E. Moreover, k

k y

-

T

E xi i=1

Y- .E (yi-yi+1) l.=1 E- y - y 1 + Nj,

Ilk~

=

y - Y1 + yk+1

j

•

Since each N. is closed. J

00

y 1 - Tx

= (y-Tx) - (y-y 1 )

Thus y 1 E- R(T) and y

E-

n j =1

N ... J

{o I

•

y-y 1 + y 1 E- R(T). Hence T is onto.

3. LIFTING COMPACT SUBSETS We have seen in prop.VI.1.5 that a linear map T is almost open if and only if every equicontinuous subset of R(T*) can be lifted by T* to an equicontinuous subset of F•. If T is a weak homomorphism, R(T*) is cr(E 1 ,E)-closed. It is enough to lift equicontinuous sets of the form

uo n

R(T*), where U is a neigh-

borhood of zero in E; these are absolutely convex and cr(E 1 ,E)compact. If F is a Mackey space, every absolutely convex and compact subset of

F~(F•,F)

is equicontinuous. So the question 113

whether a relatively weakly open map is relatively open turns out to be closely related to the possibility of lifting weakly compact sets by its adjoint. This has already been used in prop.VI.1.9,b) where the lifting problem was made obvious since T* was injective, and in prop.VI.1.10,f). Let us first investigate how, conversely, the lifting problem translates by duality. DEFINITION VI.).1. LetT be a linear map of E into F and$ (resp. $

1 )

be a collection of subsets of E (resp. F). We say

that T lifts $ such that Te

to$ if, for every e' f $

1

e

1

1 ,

there exists e f $

n R(T).

It is often sufficient to find e such that Te ) e 1 1 it suffices that en T- 1 (e 1 ) f $ i f e f $.To check this, the following lemma will be useful. LEMMA VI.).2. ~ T be a linear map of E ~F. closed in E x F, for every compact subset K

s£

li. G(T) l l

F, T

-1

K is clo-

sed in E. li. G(T) is fast sequentially closed and K fast compact, T- 1 K is fast sequentially closed in E. Let x (afU) be a generalized sequence in T- 1 K, converging 0:

to x. Since K is compact, Tx (afU) has an adherent point y f K. 0:

It is clear that (x,y) f

GTTJ =

G(T), hence Tx = y and xf T- 1 K.

For the second case, replace x (o:fU) by a fast convergent 0:

sequence xn(nflN). Then Txn(nflN) admits a fast convergent subsequence and the conclusion follows as above. PROPOSITION VI.).). ~

~

T be a linear map of E .i.!:!..:tQ F ..!D!.Q.h

G(T) is closed in E x F ~ $ ~ $' be covering families

of bounded subsets of E

~

the two following assertions (a) T lifts $

1

F, both contained in $ • Consider

•

~ $,

(b) T* is relatively open of ~(T*) equipped with the topology t$ 1 .i.!l.i£ E~.

114

~ b

~

b => a and, 1L R(T) is closed, a

~

b.

a

If b is true, given K1 T* K'

0

~

there exists

~·,

Kin~

such that

K0 n R(T*) '

)

or T*- 1 K° C K10 + ker T* • Take the polar of both sides {K 10 + ker T*)° C {T*- 1 K0

) 0

•

Observe that K1

n

RtTJ

C (K

Indeed, if x ~ K•

+ ker T*)

10

1l

iiTT') and y 1

0

•

f

K'

y2 f

0 ,

ker T* ,

Thus

Now,

(T*- 1 K0

) 0

n R(T)

Il ~ 1, ~y f T*- 1 K0

{Tx .. {Tx:

l<x,T*y>l ~ 1, ~y E- T*- 1 K 0

T(K 0 nR(T*)) 0 Since~

C

~~

}

•

fort~

and for o(E 1 ,E) are the same. But K0 is an absolutely convex and closed neighborhood of zero for t~. Thus 0 -n~R~(~T~*T) ~K~

, the closure of R(T*)

}

t~

= K0

n R(T*)

t~

(cf. the proof of (*) in prop.VI.1.5). This equality reads

___.,..........,.o(E•,E) K0 nR(T*}

= K0

li

(ker T) 0

•

Thus K + ker T •

11 5

For the last equality, observe that (K 0 n(ker T) 0 ) 0 = 'Cc;(KUker T)

=

U [eK+(1-9)ker T] eE-[o,1]

=K+

ker T

since K is compact and ker T closed. So, finally, K' n R(T) C T(K+ker T) = TK • a ~ b (when R(T) is closed). Given K 1 Eo~·, there exists K

Eo~

such that

K I n R ( T) C TK , or T

-1

K 1 C K + ker T •

Therefrom, (K+ker T)° C (T- 1 K•) 0 But K0 n R(T*) C (K+ker T) 0

•

o

Thus

K0 n R(T*) C (T- 1 K•) 0 n R(T*) • We have {T*y: yE-f>(T*) and l<x,T*y>l~1, llxE-T- 1 K 1

= T*(K 1 nR(T)) 0

}

•

Since R(T) is closed, (K'nR(T)) 0

K1° + R{T) 0

=

K1° + ker T*

o

Thus K0 n R(T*) C T* K' 0 and T* is relatively open as announced. We care now for lifting properties of compact sets. A first classical and easy result is the following. PROPOSITION VI.}.4o ~ E,F be metrizable spaces and T an open map of E onto F. 1..h.!Jl, every convergent sequence of F is the image by T of a convergent sequence of E. Moreover, 1L E is an ~-space and if ker T is closed, every compact subset of F iA the image by T of a compact subset of E.

11 6

There exist bases of neighborhoods of zero in E and F, Un (n E-lN) and Vn (n E-lN) such that TU n ) Vn , Un+ 1 C Un and Vn+ 1 C Vn for each n E-lN. Let yi(iE-lN) be a convergent sequence of F. We may assume that yi such that y i


n(y) + un + un+1 thus lj.n+ 1 (y) - q,n(y) f

2Un for each n ~ 2. Moreover, all the

q,n(y)'s belong to T-1y, which is a translate of the U-space ker T, for which U (n fiN) is a base of neighborhoods of zero. n 1 Thus the sequence ~j.n(y) (n fiN) converges to ~j.(y) f T- Y• The map q, is continuous in K. Indeed, q,n converges to

ljJ

uniformly on K s-1

s-1

~

~

i-r

i=r

Thus

Since Toq,(K)

ljJ

=

is cor.tinuous, q,(K) is compact in E. Moreover

K, hence the proof.

REMARK. The convexity of K is not preserved by the lifting, so that prop. VI.}.5 does not provide information on the lifting of

~ca

to

~ca"

Moreover, prop.VI.}.5 can hardly be used

for weak compact sets. Indeed, the assumption on ker T would 119

be that it is an ~-space for the topology induced by o(E,E•), which is the weak topology of ker T. But, if a l.c.s. E is an ~-space for the topology o(E,E 1 ) , it is isomorphic to lRw with w finite or countable. Indeed, if y n (n E-lN) E- E 1 are such that

{yi: i~n} 0 (nE-lN) is abase of neighborhoods of zero in E, it is clear that E 1 is the linear hull of {y n : n E-lN}; so its dimension is countable. Let {yi ~ i E- w) (w ClN) be a basis of E'. Every element x of E is characterized by the sequence <x,y.>(i E- w) and J : x <x,y.>(i E- w) is an isomorphism of E 1 1 intomw. The map J is surjective. Indeed, for each sequence c. (i E- w), there exists a unique linear form x on E' such that 1

Jx = c. (i E- w). Since the equicontinuous subsets of E 1 are fi1

nite-dimensional, x is continuous on each equicontinuous subset of E' for the topology o(E 1 ,E). By the Grothendieck 1 s completion theorem, x is thus an element of E. Hence the result. There is another interesting lifting theorem for webbed spaces. PROPOSITION VI.}.7. ~ T be a linear map of a subspace ~(T) ~

E onto F. If the graph of T is fast sequentially closed and

E strictly webbed. a) every fast convergent sequence of F is the image by T ~ fast convergent sequence of E, b) every fast compact subset of F is the image by T of a fast compact subset of E. a) Let ym(m E-lN) be fast convergent. It is not a restriction to assume that its limit is zero. We prove that there exists a fast convergent sequence x

m

-

0 such that Tx

m

y • m

Let B be a bounded Banach disk of F such that ym- 0 in FB. Consider the linear relation yRx Tx .. y of FB into E. Its graph is fast sequentially closed since, up to the order of the factors, it is G(T) n (ExFB). By prop. IV.6.1 (the localization property), there exists a strand 120

e

(k E-lN) of a strict web fli of E and numbers Ck \lk that

>

0 (k E-lN) such

In other words, Ck T e

\lk

) B, Vk E- IN •

Let t..k(k E-lN) be strictly associated to the strand e

(k E-lN) \lk (we may assume that t..k ) 0 by taking a subsequence) and

e:k

>

(k E-lN) be such that

0

e:k B


P

e

v

implies that

flp

o( e) \

=

u

O(e ) v

v>P

is meagre, and, for each g ~

0( e ) \

=

~

u

~'

e

e

I!

v

implies that

O(e ) v

v>I.L

is meagre. Let us prove that O(e)

(

e

u

u v~I

g

v

This shows that O(e)\ e is meagre. Given x ~O(e)\

U v~I

there exists v 1 such that x ~ O(e e

vk

x E

(k ~lN) such that x ~ O(e

k of e.

n e ~lN

E

vk

vk

v, )

g

v

and by induction a strand

) for each k ~lN. Thus

e, since the filter e

vk

converges to some point

COROLLARY VII.~.e. ~X be a l.c.s. Every absolutely convex, non meagre and Souslin subset of X is a neighborhood of zero. Let e be such a subset. By prop.VII.~.7, O(e) \ e is meagre. The conclusion follows from

prop.VII.~.6.

4. THE BOREL GRAPH THEOREM FOR L.C.S. We prove now the Borel graph theorem of L. Schwartz. The present proof is due to Martineau [72]. PROPOSITION VII.4.1.

~

E be ultrabornological, Fa Souslin

l.c.s. and T a linear map of E ~ F such that G(T) is a sgBorel subset of E x F. ~ T is continuous.

1 ~6

Assume that E is the inductive limit of the Banach spaces E (a~v). Consider the restriction T of T to E • Its graph a a a G(T) n (E xF) is a sq-Borel subset of E x F. Indeed, the set a

a

of all subsets e of E x F such that e

n

(e xF) is a sq-Borel a x F contains the sequentially closed subsets of

subset of E

a E x F and is stable under countable unions and intersections

and under complements. Thus it contains the sq-Borel subsets of E x F. If the result is proved for E (a~V), each T is continuous a a and T is thus continuous. So we may assume that E is a Banach space. We may even assume, when E is a Banach space, that it is separable. Indeed, T is continuous of E into F if it is sequentially continuous. If x (m E-lN) is a sequence of E, its closed linear hull is a sepam rable Banach space E 0 and it is enough to prove that T is continuous on E 0

•

Now, by prop.VII.2.9, a separable Banach space is a Souslin space. It i3 moreover a Baire space and we have left to prove the result when E is a Baire Souslin space. If V is a closed and absolutely convex neighborhood of zeru in F, T- 1 v

=

prE [G(T)n(ExF)]

'

where prE is the projection on E. The space E x F is a Souslin space; G(T)

n

(ExV) is a sq-Borel subset, hence another Sous-

lin space, and its projection is again a Souslin space. Moreover T- 1 v is absorbent in E, hence it is non meagre. By cor.VII.3.8, it is thus a neighborhood of zero, hence T is continuous. REMARKS. We may assume that ~(T) is non meagre in E (which implies that E is a Baire space) and conclude that ~(T)

=

E.

The result can also be formulated for linear relations and thus includes the analogous open

ma~ping

theorem. These improv-

ments are straightforward and will not be developped here.

137

The theorem of Schwartz is restrictive in the sense that all range spaces must be separable (because they are Souslin spaces). It is clear that the completing webs, by relaxing the definition of a seave, remove that restriction. We quote now another proof, due to Christensen [13], replacing the topological developments of §3 by a group theoretical argument. Let us first develop some further properties of Souslin spaces. DEFINITION VII.4o2o Let X be a Souslin space. A strict seave of X is a seave e (v~I) such that, for each strand e v

vk

( k ~lN),

00

n

e

k=1

vk

~

p •

If x is the limit of a strand e oo

n

it is clear that

k= 1

!X}

e vk

vk

(k ~lN) of a strict seave,

o

PROPOSITION VII.4.3. Every Souslin space admits a strict seave. Let e (v~I) be a seave of the Souslin space X. Define S v

the collection of all strands vk(k ~lN) of I such that v 1 and

as

v

>

v

00

e

u

v

n

vk(k ~lN) ~ Sv k=1

e vk

C e C e • Indeed, if x ~ e , there exists a strand v v v v (k ~lN) ~ S such that x ~ e for each k ~ m. The second vk vk inclusion is obvious.

We have e e

The subsets ; Indeed, let Moreover, if

X

v

~

(v ~I) define a strict seave of X. ~

I

E- e ' ~

be given. If v

>

!.It

there is a strand e

S vk

v (k

C S

1.1 ~JN)

thus e ~

s

v

C e

such

~

00

that X

~

n k=1

X

~

1 38

e

vk

Hence e

• The strand e

J.l

ev•

vk

(k

> 1)

belongs to S

v1

,

thus

~

•

Take now a strand

evk (kE-lN).

converges to the same limit as e

vk for each k , hence the result.

-

C e

X f

C e C e , it vk vk vk (kE-lN). If x is that limit,

Since e

0

vk 0

PROPOSITION VII.4.4.

Let X be a Souslin space.

a strict seave of X, each e --

1L

e (vE-I) ~ v

is a Souslin space.

I.L

-

Indeed, e (v>~.L) is a seave of e • Indeed, it is a web of e • I.L

v

x f e

vk

vk for each k, hence x f

PROPOSITION VII.4.5.

I.L

( k E-lN) converges to x, we have

More over, if the strand e

e.

be a regular Hausdorff topological

~X

space and e a Souslin subspace of X. Then e is measurable with respect to all Borel measures on X. We prove prop.VII.4.5 only for a bounded and positive Borel measure I.L on X. Let e (vE-lN) be a seave of e. We have seen in prop.VII.1.2 v

that the map

q> : nk(k flN) -

lim ec

is continuous of lNJN lim ec

n1, ••• ,nk

n 1 ' • • • 'nk

)

onto e. Moreover,

""

n

) =

k=1

for all nk ( k flN) E- lN lN • Let Ik(k flN) be finite subsets of lN. The subset K

""n k=1

is compact in

k

n

-

u

i=1 n.E-I. l.

x.

e ( n 1 ' • • • 'nk)

l.

Indeed,

00

K•

n ri

i=1

is compact in IN:N • It is thus enough to prove that

(K') C K.

Let us check the converse. Given 1 39

x

~

K,

there exists, for each k, a finite sequence

(nk, 1 , ••• ,nk,k) such that

x ~

;:;< nk ,

1 , ••• , n k , k

)'

~(k)

We can choose an element IIi

~~k) ~ l.

Since

I.

l.

~

k

-.( k

convergent subsequence n all i

~

JN 0

of K 1 such that

o

for all k

For each k and for k

~

vk

I )

~

i,

admits as

which converges to some n

large enough, we

1

~(k)

the sequence

-.( k have n 1

I )

K1 •

~

n.

l.

for

k.

Thus x~ec

and

nk

I f

1

f o o o f

nk

I f

k

I

)(e(

nk

I f

1

f • o o f

nk

I f

k

)

00

hence the result. Let now and let

£

~

>

be a

bounded and positive Borel measure on X

0 be given. Denote by

~

the set of all Borel sub-

sets of X and define A.(A)

=

inf

f~(B)

: B ~~and B ) A}

for every subset A of X.

By the theorem of Levi,

B ~ ~ such that B ) A and ~(B)

=

there exists

A.(A).

If Am T A , A.(A) ~ sup A.(Am) m Indeed,

•

choose B ~ ~ such that B ) A and ~(B ) m m m m

We may assume that Bm T.

1 40

Indeed, since

A.(A ) • m

A(Am) ~ ~(B mnB m+ 1 ) ~~(B) m

,

it follows that ~(B \ B 1 ) = o, hence we can replace B 1 by m m+ m+ Bm U Bm+ 1 and thus, by induction, obtain an increasing sequenc e B 1 ( m HN ) • Now , m 00

sup A(Am) = sup ~(Bm)

m

~(

m

U Bm) ~ A(A) m=1

,

00

because A C

U

Bm•

m=1 Since 00

e

=

U

n 1 =1 there exists thus N1 such that N1 e ) + e/2 , A(e) ~ A( U n1 n 1 =1 and, by induction, a sequence Nk(k E-lN) such that Ni A( n u e(n1, ••• ,nk) i=1 n.=1 l. k+ 1 Ni k

~

A( n

u

e(

i=1 n.=1 l.

n,, ••• ,nk+1

)) +

£

I 2 k+1 ,

lfk f lN.

Thus, by summing up, Ni

k

A(e) ~

A(i~1 n~=1

e(n,, ••• ,nk))

+e,lfkE-lN.

l.

Therefrom, if e 0 E- ~ is such that e 0 have

~ ( e 0) ~ ~ (

k

Ni

n

u

i=1 n.=1

e(n1, ••• ,nk ) )

e and ~(e 0 )

)

+

£.

A(e), we

ltk E- lN '

l.

hence, by the theorem of Levi,

~ ~(

00

k

Ni

n

n

u

k=1 i=1 n.=1 l.

e( n , •.•• , nk )) + 1

£

•

141

Since Ni

k

co

n

n

e( n 1 , ••• ,nk ) c

u

k= 1 i= 1 n.=1

c

e

l.

~·

it follows that e is measurable with respect to PROPOSITION VII.4.6.

~

X be a compact group and

~

the norma-

lized Haar measure of X. If e C X is measurable with respect

!s

~ and such

that ~(e)

>

O, ~ e

-1

.e is a neighborhood of

the unit element of X. Consider the convolution product oe

*

b _1• e

If

£

is the unit

element of X,

Since be of

£

*

o _ 1 is continuous, there exists a neighborhood w e such that oe * o _ 1 > 0 on w. Thus, for each e

y f w, oe(x) o _ 1 (yx- 1 ) is not vanishing identically e exists x f

e such that yx

-1

f

e

-1

, hence y f

e

-1

there

.e and w C e

-1

.e.

Another proof of prop.VII.4o1o By the reductions already developped, we may assume that E is a Banach space. The map T will be continuous if, for every sequence x (mflN) f E such that

l'x

m

~ 2-m,

the sequence Tx m (m flN) is bounded in F. Consider the compact group X= {0,1}lN and the continuous 11

m~~

map

co X

of X into E.

Let

~

n

be the normalized Haar measure of X. If V

is a closed neighborhood of zero in F, (To~)- 1 lin subspace of X for each n fm.

(nV) is a Sous-

Indeed, X is a separable com-

plete metric space, hence a Souslin space. The graph G(T) of T is a sq-Borel subset of E x F. Its inverse image by the continuous map (~,I) is thus a Borel subset of X x F. (Observe that

142

the class of subsets e of E x F such that (~,I)- 1 e is a Borel set contains the sequentially closed subsets of E x F and is stable under countable intersections, countable unions and complements). But this inverse image is precisely the graph of To~.

Thus G(To~) is a Souslin space and

(To~)- 1 (nV)

prX[G(To~) n (XxnV)]

is again a Souslin space.

(To~)- 1 (nV) is thus measurable with

By prop.VII.4.5, en

00

respect

U en' hence there exists n ~m n=1 _1 such that ~(en) > o. By prop.VII.4.6, en .en is then a neigh1 borhood of zero in X. Thus b mk(k ~m) ~ en .e n for m large to~·

Moreover X

enough. If b k(k ~m)

m

=

f1

m(m E-lN) - fl'm(m ~lN) , m for m

f1 1 00

00

xk =

l:

m=1

flm

X

m

l:

m=1

fl'm

X

m

-/=

k. Thus

E- 2n T- 1 v

for k large enough, hence Txk ( k ~lN) is bounded in F. 5. THE BOREL GRAPH THEOREM FOR TOPOLOGICAL GROUPS We proceed now to the extension of the results of §4 to topological groups. The vector space structure was hardly used in Martineau's proof of prop.VII.4.1; more precisely, it was just needed in order that neighborhoods of zero be absorbent and through prop.VII.3.6. A slight modification of these arguments completely avoids the vector space structure. PROPOSITION VII.5.1. 1£1 E ~ F be topological groups and T an algebraic homomorphism of E into F. Assume that E is a Baire space,

~

E

~

F are Souslin spaces and that the graph of T

is a sg-Borel subset of E x F.

~

T is continuous.

The result is still true if T is only defined on a non meagre subgroup ~(T) ~ E and yields that ~(T) is open in E. 143

Let e (vE-I) be a strict seave of F. For each v E- I, e v

v

itself is a Souslin space (prop.VII.4.4)• From ~(T)

u

T

-1

e

vE-l

v

and T

-1

e

T

v

it follows that there exists a strand e T

-1

-1

vk

e

1.1

(k E-lN) such that

e

is non meagre for each k E-m. vk Moreover, T- 1 e

=

vk

prE [G(T)n(Exe

vk

)]

is a Souslin space. By prop.VII-3.7, 0 ( T -1 e

)

vk

\ T -1 e

vk

is thus

) =/=¢, otherwise vk itself would be meagre. Thus, by prop.VII.)o5o

meagre. However, by prop.VII.).), O(T- 1 e e

vk

O(T- 1 e hence T - 1 e

).o(T- 1 e

vk

vk

)- 1

vk

.(T- 1 e

vk

)- 1 J.·s a neJ.g · hb or h oo d o f

th e unJ."t e 1 emen t

e: of E.

Since T is an algebraic homomorphism, T -1 e

vk

.(T -1 e

vk

) -1 = T -1( e

vk

.e -1) • vk

.e- 1 (kE-lN) (k E-lN) is convergent in F, hence e vk vk vk is convergent to the unit element e:' of F.

The strand e

Thus, for every neighborhood w of e:' in F, T -1 w ) T -1( e

vk

)-1 .e -1) ) O(T -1 e ). O(T -1 e vk vk vk

for k large enough, hence it is a neighborhood of e:. Therefrom T is continuous and ~(T) is a neighborhood of e:. Since it is a subgroup, it is then open in E.

144

COROLLARY VII.5.2.(28]. If a topological group X is a Baire space and a Souslin space, it is a separable complete metric space. Since X is a Souslin space, it is separable. Let e (v~I) be a strict seave of X. By the proof of prop. v

VII.5.1 applied to the identical map of X into itself, there exists a strand e (k ~lN) such that e .e- 1 (k ~JN) is a base of vk vk vk neighborhoods of E in X. Thus X is metrizable. Let now

X be

the completion of X (as a topological group).

X.

The space X is non meagre in the identical map I of domain ~(T)

Xx

X. Thus X

X into

Apply again prop.VII.5.1

to

X : it is an homomorphism, its

X is non meagre in X and its graph is closed in ~(T) is open in X. Since X is a subgroup, it

contains the connected component e

0

of

dense. So, for each connected component e open, X contains some x

~

e0

,

X. It is moreover of X, since e is

in

E

hence it contains x.e

0

e.

Comment. We may thus reduce the apparent generality of prop. VII.5.1 where the domain space E is always a separable complete metric group. we might have proved

It is true in particular for l.c.s.

prop.VII.4.1 when the domain space is an inductive limit of Baire Souslin spaces. But these are

~-spaces,

hence we are led

back to the ultrabornological case. 6. FURTHER PERMANENCE PROPERTIES OF SOUSLIN SPACES We have proved in prop.VII.5.1

(and already used in its corol-

lary) a localization property similar to prop.IV.6.1. We quote it only for l.c.s., where we intend to use it to prove new permanence properties. PROPOSITION VII.6.1.

~

E be a separable

l.c.s. and ev(v ~I) a strict seave of F.

~-space,

F a Souslin

l l T is a continuous

linear map of E into F, there exists a strand e (k ~ I) ~ -vk · T- 1 e 1"s a ne1g · h__ 1s non meagre an d that T- 1 e th a_ t T- 1 e vk vk vk 145

borhood of zero for each k e

vk

( k ~lN) with every e

~

m.

We may start the strand

such that T - 1 e

is non meagre. v1 v1 This result provides a way of exhibiting seaves in spaces

of linear maps. ~

PROPOSITION VII.6.2.

E be a separable V-space or a coun-

table inductive limit of such spaces and F a sequentially complete Souslin l.c.s. or a countable inductive limit of such spaces. ~ Lpc(E,F) is a Souslin space. Assume first that E is a separable V-space and F a sequentially complete Souslin space. Denote by Uk(k ~m) a base of neighborhoods of zero in E, by {xi : i ~lN} a countable dense subset of E and by e (v~I) a strict seave of F. v

Define the subset&(.

l.t!ltV

T ~ L(E,F) such that T- 1 e

ll

) of L(E,F) as the set of all

is non meagre, TU. C e - e and l. ll ll

Tx 1 ~e. Given&(.l.tjltV )'define&(.l.t!ltV ) t (•Jtil 1 tV 1 t~r)(ll'>ll,v'>v) V ~

as the set of all T TU. C e J

e

-

Tx 1

1 ,

&(.

l.t!ltV

~

e

1

) such that T ~

and Tx 2

-1

e .. , is non meagre, ,.

er •

ll 1 ll v ~ Continuing this procedure and relabeling at each stage by

an index in

m,

we clearly define a web in L(E,F).

A strand