E E AND WA

THE MACKEY CONVERGENCE CONDITION FOR SPACES WITH WEBS

THOMASE.GILSDORF

Department

ofPure^andApplied Mathematics Washington State University

Pullman, WA

99164 (Received December 3, 1987)

ABSTRACT. If eachsequenceconvergingto 0 in alocallyconvex space isalso Mackey convergent to0,that space is said tosatisfy theMac]eyconvergencecondition. Theproblem ofcharacterizing those locallyconvex spaces with thispropertyis still open.

In

this paper, spaces with compatible websare used toconstructboth anecessaryand asucient condition for a locallyconvexspace tosatisfy theMac]eyconvergencecondition.

KEY

WORDS AND PHHA,;ES. Compatible

web,

fast completespace, lackey convergentse- quence, inductive limit.

1980MATHEMATICS SUBJECTCLASSIFICATION CODE. Primary46A05; Secondary46A30.

1. INTRODUCTION.

In a locally convex space, it sometimes happens that every convergentsequence is also con-

vergentwith respect to a normed topologyon somesubspace. Since normed spaces have many tangible properties, it is importanttoknowfor whichlocallyconvexspaces this condition holds.

Such spaces satisfytheso-called Mackeyconvergencecondition.

Throughoutthis paper,

E

will denotea Hausdorfflocallyconvextopologicalvector space. An absolutelyconvexset in Ewill becalled a disk. If

B

is a disk in

E,

weequip its linearhull

EB

withthe semi-normedtopology generated by theMinkowski functional ofB. IfBisbounded, the Minkowski functional ofB generates a normed topology on

Es.

^If

Es

^{is aBanachspace, B is}

called aBanach disk. E isfast complete ifeach boundedset in

E

^iscontained in aboundcd

Banach disk. Everysequentially completelocallyconvex spaceisfast complete.

In Jarchow

([1], 10.1)

^{it is stated that no concise} description of locally convexspaces which satisfy the Mackey convergence conditionexists. Thisproblemis stillopen.

In

thispaperwewill examinethis characterizationproblemwithinthecontext of spaces possessing webs.

DEFINITION 1: Asequence

(x,)

^{in alocallyconvexspaceEis}Mackey convergent tox if there is aboundeddiskBC

E

suchthat

zn

^{--, zinthenorm}topology of

Ev.

^{Ifx 0,wesay}

that

(xn)

^isaMackey null

(or

^locally

null)

^sequence.

Sincethenormtopology

T

B on

EB

^{isfinerthan theinducedtopologyon}

^Ev,

^everyMackey null sequence is anullsequenceforthe original topologyon

E.

This prompts the followingdefinition ofthe converse, due to Jarchow

[1].

DEFINITION 2: A locally convexspace satisfies the Mac:key convergence condition if eachofitsnullsequences is aMackey null sequence.

Mackeyconvergence is defined by Mackey in

[2]. ^In

^DeWilde

[3] ^(Prop. III.l.10),

^{it is shown}

thatevery ^Frgchet space satisfiesthe Mackey convergence condition. Several results concerning the Mackey convergence conditionare obtained in Jarchow and Swart

[4]. In

their paper, it is shown that

E

satisfiesthe Mackey convergencecondition if and only if

T,(E’, E)

^{is a Schwartz}

topologyon

E’,

^where

T(E’,E)

^is the topology ofuniformconvergenceon all null sequences of E. Theyalso investigate theMackeyconvergenceconditionforspaces whicharefastcompleteand bornological. Specifically, thefollowingisobtained

(see [4l):

THEOREM3: Let

E

befast complete. Then the followingareequivalent:

(a)

^Eisbornological andsatisfiestheMackeyconvergence condition.

(b)

^{E ind}

line E,

^{where each}

Ea

^{is a}separable Banach space, and each nullsequence inEis null sequence insome

2. SEQ UENTIALL

Y

WEBBED SPA

CES.

Wewill nowexaminethe Mackey convergenceconditionforspaces withwebs. The readeris referred to Robertson

[5]

^fora description of websinatopologicalvector space. Further informa-

tionconcerningwebsmaybefoundinRobertson and Robertson

[6],

^{andin DeWilde}

[3].

^DeWilde

originally used websto obtainseveralgeneralizedversionsof the Closed Graph Theorem; see

[3].

In

this paperwewill workonlywith locallyconvexspaces andwewill assume, ^asin

[6],

^thateach

member ofaweb isabsolutelyconvex. The following aspectsof webswill alsobeuseful.

DEFINITION4: Astrandof aweb is acollectionof membersof

,

^{onefrom eachlayer,}

withthek

+

¹member of thestrandcontained in the the k-th member. Strandswill be denoted by

(W).

DEFINITION 5: A web on a locally convex space is compatible with

E

if for each 0-neighborhoodUin

E

andfor each strand

(Wt)

^of

,

^thereisa

k0

^ENsuchthat

Wt ^c

^U.

REMARK: Itisto be notedherethatwewillassume

(in

^{accordancewithRobertson}

[5])

^that

foreach strand

(Wt)

^{andfor each}k EN

1

(1.1)

Supposenowthat

xn

^--,0in

^E,

and suppose that forlargeenough n,

(x,)

iscontainedinsome finite collectionof strandsfrom

.

Since Definition 5implies that themembers of areinsome sensesmaller than the 0-neighborhoods in

E,

thisisactually enoughto coerce

(x)

^tobeaMackey

nullsequence. Wehavenowmotivatedthefollowing definition.

DEFINITION6: A locallyconvex space

E

issequentially webbed ifE hasacompatible web suchthatforeach null sequence

(x)

ⁱⁿ

^E,

^thereisafinite collectionofstrands,

from suchthat for each k N^thereexistsan

Nt N

such that foreachn

_ ^N,

Let usfindsomesequentiallywebbedspaces.

PROPOSITION 7: Everymetrizablelocallyconvex space is sequentiallywebbed.

PROOF" If

(Uk

^{k E}

N}

is a base ofabsolutely convex0-neighborhoods of the metrizable space

E,

such that

(Vk

E

N),

Sk+ ^c k,

1

then

{Uk"

k

N}

^{iseasilyseento beacompatiblewebon}

^E {see [5]

^or

[6]).

Certainly, every inEsuch that 0 iseventually contained inthe

{only)

strd

(U).

Thefollowing definitionmaybe foundinFloret

[71.

DEFINITION8" LetE ind

lim En

be the inductivelimitofthe locallyconvexspaces where

E1 c E2

^C and the injectionid

En

^--,

En+I

^{iscontinuousfor eachn. Then Eiscalled}

sequentiallyretractiveifeach sequence convergingin

E

isconvergent to thesamepoint in some

En.

PROPOSITION 9:

A

sequentiallyretractive inductivelimit of sequentially webbed spaces issequentially webbed.

PROOF: Let

E

ind

lin En,

^{and let}

("}

denote the webon

E,

^foreachn.

In E,

wedefine acompatibleweb as follows: Let the first layer be thecollection of all the first layers of the webs

(").

Definethe second layer of tobethe collection ofall the second layers of the webs

1(n),

andsoforth. Certainly, is acountablecollection ofabsolutelyconvexsets, andsincewe haveused all webs

(")

simultaneously, the otherproperties of webs

(as

ⁱⁿ

[6])

^{areeasilyverified}

for

.

Asfor compatibility, let

U

bea0-neighborhoodin

E,

andlet

(W)

^{beastrandfrom}

.

^Note

that

W1

^{is inthefirst}layer of

Eo,

forsomen0

N.

Since

W+I ^c W

^{for eachk,then}

W

for eachk.

Hence,

since

E

^c

E c

itfollows that foreach/E

N, Wk

^{isa memberofsome}

(i),

where1

_

^j

_ ^no.

^Ifany

^W

^E

^(f},

^{where j’}

^<

^no,then all succeeding membersof arein

E0.0),

^{since wemusthave}

Wt+l ^c Wk

^{for all k.}

^In

^fact, for large enough k, all succeeding membersof

(W)

must be in

(’0)

forsomej0; i.e., thereis a

k0

ENsuchthat

forall ^]

_ ^k0.

^{Without lossof}generality,assume

J0

^{1. Thensince U1}

E

^{is a0-}neighborhood in

E1

^{and (1) is}compatible with

E,

thereis a/

N

suchthat

W ^c UNE ^{c U.}

This makes 1/ acompatibleweb onE.

Finally,givensomen E

N,

^amember of the kthlayer of14/(n) _isalsoamemberof thekthlayer of

,

^andthisholds for eachk E

N. Hence,

each strandfrom the web

(n)

isalso astrandof

.

Nowassumeall the spaces

E

^aresequentially

webbed,

and that E is seqlentiallyretractive.

If z, 0 in

E,

then z,--*0 in some

En. Thus, (z,n)

^is contained ina finite union ofstrandsof the web

(n).

Sincetheseare also strands of

,

^itfollows that

E

issequentiallywebbed. ^[]

COROLLARY 10" Everystrict inductive limit of sequentially webbedspaces is sequentially webbed.

PROOF: Everystrict inductivelimit ofsequentially webbed spaces issequentiallyretractive sincethetopology of the inductive limit inducestheoriginal topologyon each of theconstituent spaces. []

COROLLARY 11:

Every (LF)-space

issequentiallywebbed.

3.

A

NECESSAR

Y

CONDITION.

We are nowready todescribe acollectionof locally convex spaces which satisfy the Mackey convergence condition.

THEOREM 12:

Every

sequentially webbedlocallyconvex space satisfiesthe Mackeyconver- gencecondition.

PROOF: Assume

E

^is sequentially webbed.

Let

z, 0 in E. By KSthe

[8],

28.3,

(z)

^is

a Mackey nullsequence ifand only ifthere is a sequence

(rn)

in

(0, oo)

^{such that}

r

^{cx as}

n oo, and r,z, 0 inE. We seek to find suchasequence

(rn).

^Assumethat we have found strands

(W(1)),..., (W (’))

^{fromthe web such thatforeachk Nthere isan}

N

^{Nsuch that}

(v ^_> N),

i--!

Noticethatby

(1.1),

^{foreachiand foreachk6}

^N

Hence, (’v’l e N)

wz{0

1W{0

^tz{0

1W{0

¹

W{

⁰

"k+2 k

"’k+l C _k C _k+lC

W

⁽⁰ c

1W(0

k+ 2 ^k

Fork6

N,

^wemayfind

l,

6

N

such that

Then,bydefining

wehave

W(*)

k,

+k,’

1W(t)

Wl

^{(I) C}

1-’--W(’)

C _k k,

21k,

Similarly,there is an

lk,

^6Nsuchthat

andsoforth.

Let

1

W(2)

¹

W(2)

WC)c-,_

^k, _2k,

c

lk maz{lk{

^{i 1,...,}

^m}.

Thenthereis

N

_k^{61Nsuchthatfor eachn}

^_> k

"

_,=,

^U w, ^c _,U,

¹

^w,,) . ^1w{,, .

¹

i=1 i=1

Thus,

(Vn >_ Nk)

Futhermore,thereis an

Nk+

¹

^e

^{Nsuch that}

Nk+

¹

^{> Nk,}

^and

^{(Vn >_ Nk.l)}

k+l i=1

and this continuesforallk6N.

Nowdefine

(r.)

^{by letting}

r.=k,

forN

_k

_n<Nk+

1.

lim r,, limk oo.

Itremains to beshown that

r.z.

⁰ⁱⁿ

^{E. To}

^{provethis, let Ubea}0-neighborhoodinE. By the compatibility of

W,

there arepositive integersK1,...,

K,

such that

Choosing

wehave

"’"Kin ^CU.

K rnax{K

1,...,

c Moreover,

wemay find

2’VIK ^N

^suchthatforn

^_> Nbc

by

(3.1). Hence, r..

⁰in

E,

and

(z.)

is aMackeynullsequence.

REMARK:

NoticethatbyProposition9,eachsequentiallyretractive inductive limit of sequen- tially webbedspaces satisfies the Mackey convergencecondition.

In

particular, each

(LF)-space

satisfiesthe Mackeyconvergencecondition.

If

E

ind

lim E,

^asin Definition8,then

E

^isregularifeachsetboundedin

E

iscontained in andboundedinsomeE,. Floret

[9]

^{provedthataregular}inductivelimitEof^Frgchetspacesis sequentially retractive ifandonlyif

E

satisfiestheMackeyconvergencecondition. Thisprovides uswiththe following:

COROLLARY 13" Let

E

bearegular inductivelimitofFrdchetspaces. ThenEis sequen- tially retractive if andonlyif

E

issequentially webbed. []

Inordertopresentanexample,weneed the followingfact.

LEMMA

14:

(E, 7")

^isfast complete if and only if

(E, 7")

^isfast complete, where

7"

is any topologywhichiscompatiblewiththe duality

PROOF: If B is a bounded Banach disk in

E

with respect to the topology

T,

then

B

is boundedin

E

withrespecttoany compatible topology

T’. Hence,

Bis aboundedBanach disk in Ewithrespectto

T’.

^[]

The following simple example showsthat, in additiontothe bornologicalspaces in Theorem 3,therearefast complete, non-bornologicalspaceswhichsatisfy the Mackeyconvergence condition.

EXAMPLE

15: Consider

11

^{with its}weak topology a

a(ll ,loo).

^Evidently,

11

^{under its}

normtopology

T

is bornologicai; hence,

(11,7")

is aMackeyspace, a

T,

since otherwise wouldhave finite dimension

(see [10],

page

10).

Thus,

(11 ,a)

^is notaMackey space;hence, isnotbornological.

However, (11 ,a)

^is fast complete by

Lemma

14, and the following shows that

(l, a)

issequentially webbed.

Since

(/1, T)

is a Frdchetspace, it has aneighborhood web consisting of the strand

(Uk),

^as

in Proposition 7. Furthermore, a

c T,

^so every weak 0-neighborhoodV contains a strong ^0- neighborhoodU. Thus,for the neighborhood strand

(Uk),

^{of the web}

,

^{thereis a}

k0

^ENsuch that

o ^{c c V,}

so ]# is also compatiblewith a.

Now,

if

x --

^{0 in}

^{(11 ,a),}

^then

^xn

^{0 in}

^{(/1, T)}

^{by Shur’s}

Theorem

([10],

Chapter

VII,

page

85).

That is, foreachkE

N

there isan

Nt

N such that for everyn

>_ N,

hence,

(/1, a)

issequentially webbed.

4. A SUFFICIENT CONDITION.

With some slight restrictionson

E,

wemay obtain a converseto Theorem12. Westartwitha definition.

DEFINITION 16: Atopological vector space

E

hasastrict web ifEhasacompatibleweb suchthat for each strand

(W)

^{of and eachseries}

o=

^xkwith

^{x W}

^{for each k}

^N,

MACKEY CONVERGENCE CONDITION FOR SPACES WITH WEBS 481

_ ^z

^{isconvergentin}

^E,

and further,

zE

^-1,

r=+l

foreach k

_>

2. A space withastrictweb is also calledastrictly webbed space.

If the last condition, that

isdropped, then

E

issaid tohaveacompleting web. Basically, strictly webbed spaces and^spaces with completingwebsarethose which have compatible webs and haveanadditional mild form of completeness. Thereaderis referred to

[1], I3],

^or

[5]

^formore details concerning strict webs. It shouldbe noted that in

[5]

^{strictwebsarecalled}tight webs.

There are plentyof strictly webbedspaces.

In

particular, Frdchet spaces,

(LF)-

spaces, and strong duals ofsuchspacesarestrictlywebbed;see sections 10and 11of

[5]

^forproofs.

Thedefinition of strictly webbedspaces is suggestive ofthatfor sequentially webbed spaces.

A connection between these two willbe revealedin Theorem 18 below. First, some preliminary remarksareinorder. Strictly webbedspacesalso originatedinattemptstogeneralize theClosed Graph Theorem. The most well known ofsuchresultsisusually called theLocalization Theorem;

see5.6.3of

[1]

^orsection 11 of

[5].

Wewillutilize thefollowing specialcaseofthistheorem,which islistedasCorollary2 ofTheorem 19,section 11of

[5]:

LEMMA

17: Let

E

and

F

be Hausdorfftopologicalvector spaces such that F is strictly webbed. Let

E -- ^F

^{belinearwithaclosedgraph.} Then for each bounded sequentiallyclosed disk

B

in

E,

^{thereis astrand}

(W)

ⁱⁿ

^F

^{such that foreach k GNthereexists anumber}

^a

^such

that

t(B)

^c

,w.

THEOREM 18" LetEbefastcomplete and strictlywebbed. IfE satisfies the Mackeycon- vergencecondition, then

E

issequentiallywebbed.

PROOF" Let beastrictwebon E.

Let

z, ^-, 0inE. By assumption,

(zn)

^{is a Mackey}

null sequence; therefore, by KSthe

[8],

^{28.3, thereis a sequence}

(r,)

^c

(0, x),

^{with r, c as}

n xandr,,x,, 0inE. Let

A

{r.z.

^:n

N}.

ThenA isbounded,soit iscontainedinabounded BanachdiskB. Furthermore,

A

is bounded inthe Banachspace

Ez

and by letting Cdenote the

Ez-elosure

^{ofthe convex,balancedhull of}

A,

then clearly C is a bounded, sequentially dosed disk in

En.

^Notice also that the injection id

En

^Eiscontinuous, therefore ithasadosed graph. ItfollowsnowbyLemma17 that there existsastrand

(Wt)

^of such that for everykE

N

thereisanumberat suchthat

ia(c) c c ,w;

therefore,foreverynE

N,

r.x.

E

atWk.

Since

rn

^{oo asn} o, for^afixedk Nthereis an

Nk

^{Nsuchthat forevery n}

^{_> N,}

Thus,foreveryn

> N

Iw ^cw,

since we assumethateach

W

^{isbalanced. []}

As was mentioned before Lemma 17, the strong dual ofa Frdchet space is strictly webbed.

Moreover,

by Corollary^{2 of}Proposition 1, ChapterVIof

[6],

the strong dual ofaFrdchetspace is complete, hencefastcomplete. Thisallowsus tomake the following statement.

COROLLARY 19: The strong dual of a Frdehet space satisfies the Mackey convergence condition ifand only if it issequentially webbed. []

ACKNOWLEDGEMENT. ThisresearchwassupportedbyWashingtonStateUniversity.

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