INCOMPLETE REGULAR INDUCTIVE LIMIT OF BANACH SPACES

Internat. J. Math. & Math. Sci.

VOL. 13 NO. 4 (1990 817-820

817

EXAMPLE OF A

SEQUENTIALLY

INCOMPLETE REGULAR INDUCTIVE LIMIT OF BANACH SPACES

JAN KUCERAandKELLY McKENNON

Department

of Mathematics WashingtonStateUniversity Pullman,Washington 99164 (Received August ^22, 1989)

ABSTRACT.

A

sequentially incomplete regularinductive limitofasequence of Banachspaces is constructed.

KEY WORDS AND PHRASES. Regular locallyconvex inductive limit, sequentially complete locallyconvexspace.

1980MATHEMATICSSUBJECT CLASSIFICATION CODE

(1985 Revision):

^{Primary46A12,}

secondary46M40.

I. INTRODUCTION.

In

[I]

^JorgeMujica asks: Iseveryregularinductive limitof Banachspacescomplete? Apartial answer is in

[2]

with anexample ofa regular inductive limit which is quasi-incomplete. It is conjectured in

[2]

^that the regularinductivelhnitmight evenbe sequentially incomplete. Here

weprove the conjecture. On the otherhand,regularinductive limits ofBanachspaces always havesomecompleteness property,e.g. theyarefastcomplete,see

[3].

2. MAIN RESULT.

Let N

{I;2,...}

^andRbe the space of real numbers. Foreach z

{zj}

ER^NxNand nEN

weput z

I[,.= max{, ^{max{Izj[;] <} n},sup{Izj[; ] >_ n}},

Forbrevity,wealsoput

LEMMA

I. The mapz

II = II.: . _- ^.

^{norm on}

^E,

and each functional]’j:z

-

^zj

E ^R

is continuous.

818 JAN KUCERA AND KELLY McKENNON

LEMMA

2. Each

E.

^isBanach.

PROOF.

Let

{xC);k }

^beCauchy. Thenforeach i,j E

N,

thesequence

{J’d,iCz(k));k N}

Cauchyin

R

andthus convergesto_{some x,i} R. Put z

{z,i}.

^Foranyi,k,n

^N,

^we

have lim

z(k) z(m) II,.=11 () I1,.,

^{whichimpliesz}

E

^{d lira}

z(k)

^z

I1=

^0.

indlim

LEMMA

3. Theinductive limit

E

n oo, is regular.

PROOF. Let BCEbenotboundedin any

E,.

Withoutaloss of generalitywemay assume that foranyn 5N thereexists

x(n) B

suchthat

x{n) I1,>

^{n. This}implies the existence of sequences

{i(n)}, {j(n)}

CNsuchthat: either

j(n) >_

ⁿ and

x(n)q},i{,} I>

^{n orj(n)}

^<

^{n and}

(),l. !> ⁱ⁽⁾ .,.

Foreach k

N,

choose

r >

^{0 sothat}

r; max{i(n);

ⁿ

_ ^k}

^{anddenotebyVtheconvex}

hull of

U{E(r);k N).

^Takek,n

N,x {x,.} E(r),

and distinguishthreecases:

(a) jCn) >

k,whichimplies _zq,la.{,}

i_<ii

^z

II,< ,, < _{< ;} z(n),.l,. l, (b) i(n) <

^k

^&

^k

>_

n,whichimplies

-" (-),(,)..,,)

(c) j(n) <

^k

&

k

<

n,whichimplies _xi{,}.j{,}

I_ i(n). I1_< i(n)r _ ^{i(n) <}

(,’,,),.,,,, I.

Fo ,., I<

^g

(),c-a. I.

^{Since x}

^E(r),k

⁵

^N,

^{was arbitrary, the}

element

-xCn)

^cannot be expressed as a convex combination of elements from

U{E(r);k e N},

i.e.

x(n)

V.

Hence

the 0-neighborhoodVin

E

doesnotabsorb

B

^and

B

isnotbounded inE.

LEMMA

^4.

For

each i,j,n

N,

put

x(n),

d j-if

<_ n,x(n),

d 0if i

>

n, and

x(n) {x(n)id}.

^Then:

Ca)

^Foreh.

e N, zCn) e El(1).

() ((,)} c

ⁱ

.

(c) {zC)}

^{does not}

_converse m

E.

PROOF.

Ca)

^isevident.

Cb)

^{Let V bea}neighborhoodin E.

For

eh

N,

choose

r. >

^{0so that}

E.(r.)

C V.

Fther,choose p, q Nothat W

>

2dqr

>

2. Form,n

>

q, definey, z

E

by:

yi..

xCrn)id xCn)id

^{for j}

^_>

p, Yid 0otherwise,

z. ^x(rn),

d

x(n),

d for j

<

p,

z,

⁰otherwise.

REGULAR INDUCTIVE LIMIT OF A SEQUENCE OF BANACH SPACES 819

Since

O for

I=(),, =(),, I<_ -

^{1 forfor i}i>q,j<p

^>

^q,j

^>

^p Wehave

v I1 ^sup{I

^v,j

^{I;i e}

^,i

^>

(c)

Assume

z(n)

^{z in}E. ^Sinceeach functional fj, defined in Lemma 1, is continuous

o=

= .,

^{it Io =o=tin,o,on}

. ^Z=

^w

^{e ,. f,.() limf,.(())}

lim

zCn),# -I.

^{Hence z}

Ill,n= max(7, )

^{and lim z}

I[,.=

^{0 i.e. z}

E

^forany

n.N.

BycombiningLemmato1-4,weget:

THEOREM.Regularinductive limit ofBanachspacesmay be sequentially incomplete.

REFERENCES

[1] MUJICA, J.,

Functional Analysis, Holomorphy andApproximationTheory

II,

North Hol- land1984.

[2] KUCERA, J., MCKENNON, K.,

Completeness of regular inductive limits,Int. J.Math.

&

MathSci.Vol. 12,No. 3

C1989),

^425-428.

[3] ^DEWILDE, M.,

Closed Graph Theorems and WebbedSpaces,Pitman, London 1978.