BOUNDED SETS IN _/’(E,F)

Internat. J. Math. & Math. Sci.

VOL. 12 NO. 3 (1989) 447-450

447

BOUNDED SETS IN _/’(E,F)

THOMASE.GILSDORF

DepartmentofPureand AppliedMathematics WashingtonStateUniversity

Pullman,Washington 99164

(Received October 27, 1987 and in revised form June 7, 1988)

Abstract: Let

E

and

F

be Hausdorff locally convexspaces, and let

(E,F)

^{denote the space of}

continuouslinear mapsfromEtoF. Suppose that for everysubspaceN c

E

andan absolutely convexsetACEwhich isbounded, closed,and absorbingin

N,

thereisabarrelDC Esuchthat A

D

^{fN. Then}it isshown that thefamiliesofweakly and strongly bounded subsetsof

(E, F)

areidentical ifand onlyifEislocally barreled.

Key Word and Phrases: Locally barreled space,S-topology,bounded set for S-topology.

1980MhernaHco Subject

Glassifieation

^Code

(185 Rer4oion):

^{Primary40El0;}Secondary 46A05.

I. INTRODUCTION.

Throughoutthispaper

E

andFwill denote Hausdorff locallyconvexspaces,and

(E,F)

^the

space ofcontinuouslinear mapsfrom

E

toF.

An

absolutelyconvexset B in

E

will becalled a disk. IfA is anysubset of

E,

^itslinearhull will be denotedby

Ea.

^Foradisk

^B

ⁱⁿ

^E,

^{its linear}

hullisgiven by

Es U{nB

ⁿ

^>_ 1}.

^Equippedwiththe topology generated bythe Minkowski functional of

B, EB

^{is a}semi-normed space. Thisleads tothe definition whichfollows.

DEFINITION1: Let

B

C E be a disk. If

Es

^{is a}barreled normedspace, then B is called a barreled disk;

E

^islocally barreled ifeachbounded set in

E

is contained in a closed, bounded barreleddisk.

448 T.E. GILSDORF

II. A UNIFORM BOUNDEDNESS THEOREM

Itisproven in

[11

^{thatinalocallyconvexspace}

^E

^thefamilies of

a(E’, E)-bounded and/(E’, E)-

bounded sets arethesameifEis locally barreled. This is proven for thegeneralcase

L(E, F)

ⁱⁿ

ourfirst result below. Conversely, insection III. wewill examine the local barrelednessofE in termsof subsets of

(E, F)

whichareboundedfor anyS-topology,where

S

isafamily ofbounded sets which coversE.

THEOREM"2. If

E

is locally barreled then the families ofbounded sets in

..(E,F)

^{are the}

sameforall S-topologies, where

S

^is afamily of boundedsets in

E

whichcoversE.

:PROOF:AssumeEto belocally barreled. Let Vbeaclosed,^{absolutelyconvex}0-neighborhood inF. Let H c

(E, F)

be pointwisebounded. Let

D

{u-’(V):u ^e

ThenDisaclosed disk inE. SinceHispointwisebounded,wehave:

forsome a

>

^{0. By}taking inverse images,itfollowsthatDisabsorbing in

E;

hence,

D

is abarrel in E. In 8.5, ChapterIIof

[2]

it is proven that Dabsorbsall bounded Banach disks. A careful readingofthatproofreveals thattheonly propertyofBanachspaces which isusedisthe property of beingbarreled.

Hence,

anybarrelinEabsorbs allclosed,bounded barreleddisks in

E,

^aswell.

Moreover,

ifAis anyboundedsubset of

E,

thenA iscontained insomeclosed,boundedbarreled diskB. Therefore, DabsorbsA and 3.3, ChapterIIIof

[2]

^nowasserts thatHisboundedforthe topologyofboundedconvergenceon

(E, F).

^[]

Ill. LOCALLY

BARRELED

SPACES AND BOUNDED SETS IN

f.(E,F)

Let

(P)

denote the following property ofalocallyconvexspaceE:

(P)

^Foreach absolutely convex,closed,bounded setAC

E

thereexists a barrelD c Esuchthat A

D

^q

EA.

THEOREM8: LetEandFbeaHausdorfflocallyconvexspaces. AssumeEsatisfiesproperty

(P).

^{Then the}followingareequivalent:

BOUNDED SETS 449

Thefamiliesofbounded subsetsof

.(E,F)

^{areidenticalfor all} S-topologieson

..(E,F),

where $ is afamily of bounded subsetsofEwhichcoversE.

(b)

^Eis locally barreled.

PROOF.

In

viewof Theorem 2,weneed onlyprove

(a) = ^(b).

If

E

^isnotlocallybarreled,thenthereexists anabsolutely convex,closed,bounded setB c Esuch that

EB

^is not barreled. Wewill firstshowthateveryset

M

which isclosed and boundedin

EB

isalso closedinE. Denoteby

M0

the closure of

M

inE. Since

M

isboundedin

E, M c AB,

for some

A >

0.

AB

isclosedinE. Hence

M0 c ABc E.

^Takex0in

M0

^andanetr/cMsuch that

r/ x0inthe topology ofE. The identityid

E

^Eis continuous, and

{k-lB

k E

N}

is abasisfor the neighborhoods ofzero in

E

consisting ofsetsclosed inE. Therefore,^by3.2.4 of

[3],

r} x0in thetopology of

EB.

^{Finally,Mis closedin}

E.

^Hencex0E

M,

soMisclosedinE.

Nowchoose a barrel

A

in

E

^{which is not a} 0-neighborhoodin

E.

^{Then wemay choose a}

sequence

(xn}

^C

EB\A

^{such that}

xn

⁰ⁱⁿthe topology of

E.

^Thenormabilityof

EB

^implies

that

(xn)

islocally convergent; thuswe maychooseasequence

{an}

^{ofpositiverealnumberssuch}

that

an

^{oo and}anXn 0 inthe normed space

E.

^Sincethe normed topologyof

E

^{is finer}

than the topologyon

E

^inducedby

^E,

^thesequence

(anxn)

^alsoconvergesto0withrespect to the topology ofE. This means

isboundedinE.

SinceAN

B

is absolutely convex,bounded,andclosedin

EB,

it isalso closed and bounded in E. By

(P),

^thereis abarrelDC

E

such that

A n B D n E,an D n EB

Now,

_Xn

D

for each n, andwemaytherefore choose

fn ^E’

^{such that}

I/.()1 <

^{forany x6D}

while

fn(xn)

^{1, whereeach}

fn

^{isrealvalued.}

Let y0

F\{0},

^{and defineg:R Fby}

(z)

0,

foreachz R. g isalinear map taking bounded sets inR tobounded setsin

F;

therefore, g is continuous.

450 T.E. GILSDORF

Now,

for eachn 6

N,

define

hn ^E

^Fby

h.

^go

f..

Asthe compositionoftwolinear,continuous maps, eachh,

(E, F)

Put

H={h.’n6N}.

First,noticethatfor each x

D, If,(x)l _<

1, hence

h,(x) C,

whereC isthe linesegment from-y0toy0inF. Obviously,Cis boundedin

F;

consequently,

U{h.(z) - ^{e N}}

isbounded inFfor each x6D. SinceDis absorbing in

E,

isbounded inFforeach x6E aswell;thismakesH apointwiseboundedset.

Finally,

U{h.(x)’x ^{e S,.}

⁶

N} U{h.(a.x.)’n e N} U{a.g(1) .

⁶

^{N} U{a.{yo}} . ^{e N}.}

Letting

a. a*,

^then

while

lira

a.

^0,

lira

.(a.{yo})

Yo

#

^0.

Thismeans

H(S)

^{mnotboundedin}

F;

^thusHisnotbounded for the topology ofuniformconver- genceonbounded sets. []

Present address:

Department of Mathematics and Computer ^Science

University of Wisconsin

River Falls, WI 54022 References

1

KU(ERA, J.,

GILSDORF,

T.,

A Necessaryand Suificient Condition forWeakly BoundedSets tobe StronglyBounded,toappear.

SCHAEFER, H.,

TopologicalVector Spaces, Springer, 1971.

3 JARCHOW,

H.,

Locally ConvexSpaces, B. G. Teubner Stuttgart, 1981.