THE STRONG WCD PROPERTY FOR BANACH SPACES

Internat. J. Flath. & Flath. Sci.

VOL. 18 NO. (1995) 67-70

67

THE STRONG WCD PROPERTY FOR BANACH SPACES

DAVE WILKINS

(Received June

^17,

1993)

ABSTRACT. In

this paper, ^we introduce weakly compact version of the weakly countablv determined

(WCD)

property, the strong

WCD (SWCD)

property.

A

Banach space X s said to be

SWCD

if there s a sequence (A,) ^{of weak.} compact subsets of

X**

such that if KCX is weakly compact, thereis an (n,)N such that h’C

= ^1An

^CX.

^In

^{this case, (A,) is calleda}

strongly determining sequencefor

x. We

show that

SWCG SWCD

andthat theconversedoes nothold in

general. In

fact,

x

is a

separable SWCD

space if and

only

if (X,

weak)

isan0-space.

Using _o for an example, we show how weakly compact structure theorems may be used to construct

strongly determining

sequences.

KEY WORDS AND PHRASES.

Banachspaces,

WCG, WCD, SWCG.

1991

AMS SUBJECT CLASSIFICATION CODES. 46B,

54C.

1.

INTRODUCTION.

Let

X be a Banach space with dual space

x*

and second dual

x** Let

B and B** denote theclosed unit ballsof

x

^and

X**

respectively.

Xis said tobe

weakly compactly generated (WCG)

^{ifthereis a}weakly compact KCX with thespan of Kdensein X

[3].

^The

^WCG

^{propertyhas beenan active}topic of research for several years

(e.g., ([1], [3], [8]).

^{Similarly, a} generalization of this property, the

WCD

property, has been investigated,

pa:ticularly

since

WCD

spaces possess many ofthe sameproperties as

WCG

spaces

(e.g., [6], [11], [12]). x

is said to be

WCD

ifthere is a sequence

(A,)

^ofweak. compact subsets of

X**

such thatfor each zeX thereisan

(n) c

^{Nwith e}

f’l= ^A,

^CX

[12]. In

this case, we say that

(A,)

weakly determines X.

We

will see that each ofthese properties may be expressed as a property of the family of norm compact subsets of

x. Our goal

here is to introduce the

weakly

compact version of the

WCD

property, the

SWCD

property, and to examine itsrelationshiptothe strong

WCG

property of Schltichtermann and Wheeler

[9].

We

firststatesomedefinitionsand results.

X is

strongly WCG (SWCG)

^{if thereisasequence}

(K,,)

ofweakly compactsubsets of Xsuch that for each

weakly

compact subset H of X and each >0, there is an hen such that HC

K, +

^B

[9]. ^As

^noted in

[9],

restrictingH tonormcompact sets intheabove definitiongives

adefinitionof

WCG

thatisequivalent totheoneabove.

x

is

SWCD

ifthere is a sequence

(A,)

^ofweak. compact subsets of

X**

such that for each

weakly

compact KCX there is an

(n,)c

^{N with KC}

f’]= 1An

^CX.

^In

^{this case, we say}

^(An) strongly

determinesX.

The following result affirms the claim that

SWCD

is the natural definition for the weakly compact versionof

WCD.

B8 D. ^ILKI_NS

PROPOSITION

1.

x

is

WCD

if and oulv if there is a sequence (..1,,) of weak.

compact

subsets of X** such that for each noru (’Oral)act K X, there is an (,,,,)cN with

Uc

N=A, cx.

PROOF.

()Clear.

() Suppose

_(C,) is a sequence of weak* compact subsets of

x**

that veakly determines

x.

B**

Now

let (A,) ^bean enumerationof the finite unions of the

For

each

,

jE N let

F,,

’ ⁺⁷

F,,,

^{andnote that each}

A,,

^isweak compact.

Suppose

K is a norm compact subset of X. Choose a sequence

(n)C

^{N so that}

E(nn)K^C

A,. We

^{certainlyhave K}C

N .4,,

^{so weneed}

^only

^{show that}

M A,

^CX.

Le **E X**kX. ^For

^{each zEK there is an ()EN such that}

EC,()

^and

**

B**.

Note

that this lmst set h

nonempy

There is also a j() E N such that

** ^C,(.) +

norminerior, so, infac,wemay find

,...,.

^{K suchthat}

=1 and

...)

he

se

on the right is one of the

A

containing K, hence it is one of the

A

^{so we have}

** ^{8= A,} .

We

have

WCGWCD

from

[12],

^{and the}

analogous

result for the

stronger

properties

from

the following.

PROPOSITION

2. IfX is

SWCG

then

x

^is

SWCD.

B**

wih j,n N, where PROOf.

Let

the

(a,)

^be enumeration of setsof the form nK

+

K isan

SWCG generator

forX.B

I

iswell-known that

WCD

spaceseLindel6f inhe

we opology [12]. Utilizing

strengthening of theLindel6fproperty,wehaveasimilresult for

SWCD

spaces.

A

family ofsubsets ofa opologicspace T is called astron₉ open coverof T, if open coverofT d for each

compac

subset K

o

T thereis aU6 with KCU.

If

every

strong

open cover of T has a countable

strong

open

subcover,

T is sd to be

strongl Lindel6f

^(SL).

Thispropertyw first studied in

([4], [5])

in relationtopropertiesofthe

compact-open

onspacesofcontinuous functions.

PROPOSITION

3. IfX is

SWCD

then (x,

weak)

^is

^strongly

^Lindel6f.

PROOf. Suppose (A,)

^{is a}sequence of

we.

compact subsets ofX’*

strongly

determining X, d let

{U}

^bea

^strong

^{open cover} of (X,

wea). or

^{each 6}1 there is n

such that

V

^{is w.open d}

v ⁿ

^X

U.

Le

^{q be} the collection of

1

finite

subsequences, ,

^ofN such that

, ^,A, V ^for

^some

a6 1. Infinitelymy such exist sincethereeinfinitelymy

(n) c

^with

A

^X.

We

maysume q {i}=a.

Eor

each 6 chseu,⁶¹such hat

,,Aj

^C

Le

K be a wetly compact subse of X.

By

hypothesis there is

(n) c

^{such that} K

= ^aAn

^{X. There is also a6 1 such that KC}

= ^An

^C

^U

^C

^{V, hence,}

^since

V ^c

^{X** is}

we.

open, hereis a 6 such that KC

’, A,

^C

^{V. Now}

^{m, .,m i}

^o

some i, so KC

Vai.

^{Since KCX, wehave KC}

U,. ^herefore, (U,)

^{is a countable}

^strong

^open

subcover of.B

hek of

identifying SWCD

spaces maybereduced

by

heorem1.

In

orde to prove this

esult,

wefecal he following definition.

Let

Tbe a

completely rel topological

space,d let 9beafily of subsets ofT. 9is

STRONG WCI) PROPERLY FOR BAN,\CH SPACt.S 69

saidto l, a l"doba" f ^7"iffl ^’,(hopen c"1 and each conpact !c thereis a such that KCl’C l" If "I has a tntallepse(lol,,e. I" is said to t, _an

Ro-sIace [7]. ^A

^{recent study}

ofs0-space, in

regard

to Banachspaces ^i,giw’n in

[10]. ^In [7]

it i,prv,,,1 that fT isan R0-space then T is Lindelaf,and, in fact,an eh’mentarv modification of thisproofreveals that "I isstrongly Lindel6f.

The followingesult indicateb that sepatallityprovides away t "’isolate"" a w’aklv compact convex set K flora

XK

^ing intersection of ^menl)er of a countal)h’ fanlv of weak. compact subsets of X’" The

SWCD

propertyis the (ondition needed toisolate A flown

.V’*X.V.

will beanN0-space precisely when ^{K is isolat(’d}from

X"K

^{in thismanner.}

LEMMA

^1. Let X be separable. Then the’reisa sequence(F,)^of

,.,

compact subsets of such that if K is a weakly compact ^convex subset of X, there is an (,,,)c ^with K

N :

^=,

F. )Nx.

PROOf. Let the norm on

x

be denoted by

1. I. ^From [1]

^{there is an equivalent norm.}

II1-111

^on

^x,

^{uch thai every} weakly compact convex subset of X can be written as the intersection of closed

II1" II-bls.

Now

let (,)be adense sequence of points in {x,

lll-IIl. ^Suppose

^{that K isaweaklycompact}

convexsubset of X and

EX.

^{Let zEx and a>0 be such that KCB(z,b)} and B(z,b), where theball istheclosed ball withrespect to

II1 Ill.

We clearly"may

enlarge

this closedbll

o

rdius so that

r-a

>0, is rational, and B(,,,.).

Set =,,{{-a),lll-lll-),

and find nsothat

II1-111

^<

.

^{Then (z,p)DK. This shows that for X separable, it, is}

^enough

^{to consider}

those closed balls centered at some

z

and of rational radius in the previousparagraph. Let the

(Fn)

^beanenumerationofthe m,closures in A"* of this collection ofclosed

III. Il-b.

THBOM

1. IfX isseparable,thefollowingareequivalent.

()

^xisSWCD.

()

^(X,

wea)

^{is n}R0-spce.

()

^{There is a} sequence

(A,)

^{of m,} compact subsets of X’" such that if K is a

weakly

compact^convexsubset of

x,

then thereis an

(n=)

^{c with K}

n ^A, _m-

PROOf. (12).

Suppose

_(An) is sequence of ^m, compact subsets of X*" that strongly determines X. Choose the sequence (F,,) according ^to

Lemma

and let (U,) denote ⁿ enumerationof the members of

(A,)

^and

(F).

^Thenlet

’= ^(P)

^{be asequence}formed from 11 finite intersectionsof members of(C),^ndset

(P,)

^where

P, P

^{NXfor ech E}

.

By

the sub-base theorem in

[7],

^{it is}

enough

^to show that for ech

weakly

open ^convex set UcX nd wekly compact ^convex set KCU, thereis n nE such hat

KCP,

CU.

Assume

d K regiven this way, where

u=vnx

for somem,open Vcx*’. Then thereis

(nm)

C

such ht K=

n=u ^Hence

^hereis E such that

n m=U. ^cv,

^but

^n=G, =P’

for somej, so

KcPcV.

^Thus

KCPCU.

(2a).

Assume

h& _(X,

wea)

h&8 & countable

pseudobse, =(P).

^{Without loss of}

generlity

ssumethat ech memberof is bounded nd wekly closed in

x. or

ech let A

c

x’* and notethat ech

A,

^is

compac

in

Let

KCX bewekly compact, and choose

(n) c

^{so that} E

(n)K

^C

A,. Suppose X**K.

^{Thenthereis m,open set V}

c

x*" such that KCVnd

*" ^g

^’.

^By

hypothesis, there is an nE such that

KCP,

^{CV, so}

KC’CV’,

nd hence KCA nd

,’*A,.

^Therefore

= ^{N= a,}

(al). Obvious.

It

should be noted that

SWCD

ds not imply separability, since 11 reflexive spces are

SWGD,

nd separability ds not imply

SWCD,

because there exist separable Bnch spces,

70 I). WILKINS

C([0,1]) for instance, which are not 0-space in the weak

topology [7]. ^A

^simple exainple ofan

SWCD

space that isnot

SWCG

isgiveninthefollowing.

EXAMPLE.

Since _o is not weakly sequentially oomph’re it cannot be

SWCG [9]. However,

(c

o,weal:)

^is

an 0-space,since it hasseparalle dual

[7].

^so

o

^is

^SWCD.

^Thefollowing example demonstrates how strongly determining sequences maybe produced by utilizing results ^al)out the structure of weaklycompact sets.

From [2]

^weobtainthe following result.

Let

M_{C o.} Then M is relatively weakly compact if andonly if M isbounded and forevery (rnt)

c

^Nwehave

. ^m:

^supM

(

^in,1,

^Xrnk 1)--,0,

Let

e

bethe collection of all finitesubsequencesofN.

For

n, Nand

e,

set

Then each A

....

^{is bounded and w. closed, hence w, compact. The collection of all A}

....

^is

countable,

soletC beanenumerationof the A

’

^rit"

Suppose

K is a

weakly

compact subsetof _0.

Let (m)c

^{Nbe the}collectionof all n hrsuch that KC

C,,

noting that thereareindeedinfinitelymanysuch

C,, by

^theabove result.

Now

suppose z’*6

e\c

0. Then there is a j 6/vand a (t)c/v ^{such that}

z;’

^{>1/j for all}

k>_1.

By

the above result

[2]

^{again, KCA}

....

^{forsome 6h"and rofthe form r t,t,.}

^.,t,,

yet z*" iscontained in nosetofthisform. Thus z**

i"1= 1C,,.

^Therefore

^f’l= C, ^c

^{X, hence}

(C,)

is a

strongly

determiningsequencefor _0.

REFERENCES

1.

AMIR, D.,

and

LINDENSTRAUSS, J.,

The structure of

weakly compact

sets in Banach spaces,Annals

of

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(1968),

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BATT, J.

and

HIERMEYER, W., On

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topology

and inthe

topology a(L,(t,X),Lq(t,X’)),

^Math.

^Z.

¹⁸²

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3.

LINDENSTRAUSS, J., Weakly

compact sets their

topological

properties and the Banach spacesthey

generate, Ann. of

Math. Studies 69

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4.

MCCOY, R.,

Function spaces whichare

k-spaces, Topology

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5.

MCCOY, R.,

k-spacefunction spaces,

Inter’nat. J.

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6.

MERCOURAKIS, S., On weakly

countablydeterminedBanachspaces,

Trans. Amer.

Math.

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300

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7.

MICHAEL, E.,

tl0-spaces

J.

Math. Mech. 15

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8.

ROSENTHAL, H.,

The heredity problem for weakly

compactly generated

Banach spaces,

C..omp.

Math. 28

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9.

SCHLUCHTERMANN, G.

and

WHEELER, R., On strongly WCG

Banach spaces, Math.

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10.

SCHLOCHTERMANN, G.

and

WHEELER, R.,

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Mackey

dual ofaBanach space,

Note Mat.

MemorialVolume for

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VALDIVIA, M., Some

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12.

VASAK, L., On

one generalization ofweakly compactly

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Math.

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Mathematical Problems in Engineering

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

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