SEQUENTIAL COMPLETENESS OF INDUCTIVE LIMITS

Internat. J. Math. & Math. Sci.

Vol. 24, No. 6 (2000) 419–421 S0161171200003744

© Hindawi Publishing Corp.

SEQUENTIAL COMPLETENESS OF INDUCTIVE LIMITS

CLAUDIA GÓMEZ and JAN KUˇCERA (Received 28 July 1999)

Abstract.A regular inductive limit of sequentially complete spaces is sequentially com- plete. For the converse of this theorem we have a weaker result: if indEnis sequentially complete inductive limit, and each constituent spaceEnis closed in indEn, then indEnis α-regular.

Keywords and phrases. Regularity,α-regularity, sequential completeness of locally convex inductive limits.

2000 Mathematics Subject Classification. Primary 46A13; Secondary 46A30.

1. Introduction. In [2] Kuˇcera proves that an LF-space is regular if and only if it is sequentially complete. In [1] Bosch and Kuˇcera prove the equivalence of regularity and sequential completeness for bornivorously webbed spaces. It is natural to ask: can this result be extended to arbitrary sequentially complete spaces? We give a partial answer to this question in this paper.

2. Definitions. Throughout the paper{(En,τn)}n∈N is an increasing sequence of Hausdorff locally convex spaces with continuous identity maps id :En→En+1 and E=indEnits inductive limit.

A topological vector space E is sequentially complete if every Cauchy sequence {xn} ⊂Eis convergent to an elementx∈E.

An inductive limit indEn is regular, resp.α-regular, if each of its bounded sets is bounded, resp. contained, in some constituent spaceEn.

3. Main results

Theorem3.1. Let each space(En,τn)be sequentially complete andindEnregular.

ThenindEnis sequentially complete.

Proof. Let{xk}be a Cauchy sequence in indEn. Then the set

{xk; k∈N}is bounded in indEnand since the inductive limit is regular, we may assume that it is also bounded inE1. For everyn∈N, denote byPntheτ1-closure of the convex hull of

{xk; k > n}. LetQbeτ1-closed balanced hull ofP0andF the span ofQwith the topology generated by the filter basis{λQ; λ >0}. Denote this topology byµ.

Before we finish the proof, we prove two lemmas.

Lemma3.2. (F,µ)is Banach.

Proof. Let{yk}be a Cauchy sequence inF. Since the setQis bounded inE1, each µ-neighborhoodλQis absorbed by anyτ1-neighborhood of zero inE1. This implies that the topologyµ is finer than the topology ofF inherited fromE1. Hence{yk}is Cauchy inE1and as such, it converges to somey∈E1in the topologyτ1.

420 C. Gómez AND J. Kuˇcera

For anyµ-neighborhoodλQof zero, there existsk∈Nsuch thatyp−yq∈λQfor anyp,q > k. If we letq→ ∞, theτ1-closedness ofλQimplies thatyp−y∈λQfor anyp > k, that isy∈yp+λQ⊂F andyp→yin the topologyµ.

Lemma3.3. The respective families ofτ-bounded andµ-bounded sets in the space (F,µ)are the same.

Proof. Since the topologyµ is finer thanτ1, which in turn is finer thanτ, any µ-bounded set isτ-bounded.

LetA⊂Fbe aτ-bounded set. Denote byBtheµ-closure of the balanced convex hull ofA. PutG=

{nB; n∈N}and equip it with the topology, which we denote byγ, generated by the filter basis{λB; λ >0}. It follows from Lemma 3.2, that(G,γ)is a Banach space.

LetP0andQbe the same sets as above, we first show that the setQ

Gis closed in(G,γ). Let a sequence{zk}⊂Q

Gbe convergent to an elementz∈Gwith respect to the topologyγ. Sinceγis finer thanτ1, zk→zalso in the topologyτ1and sinceQ isτ1-closed, we havez∈Qandz∈Q

G. Similarly, the setsQn=n(Q

G), n∈N, are closed in(G,γ)andG=

{Qn;n∈N}. By Baire’s category theorem, there exists n∈Nsuch thatQnhas nonempty interior andQn−Qn=Q2n is aγ-neighborhood of zero. SinceBisγ-bounded andQ2nis aγ-neighborhood of zero, thus there exists β >0 such thatB⊂βQ2n⊂β2nQ, that is,Bisµ-bounded.

To continue the proof of the theorem, we observe that the weak topologyσ onF is the weakest topology onF for which the family of allσ-bounded sets inF is the same as the family of allµ-bounded sets inF. So we haveµ⊃τ⊃σ and theτ-Cauchy sequence{xk}is alsoσ-Cauchy.

LetFbe the strong second dual ofF. ThenFcan be considered as a closed subspace ofF. Hence eachf∈Fcan be continuously extended toFand theµ-closed convex set P0⊂F is alsoσ (F,F)-closed inF. Moreover,P0 as a set bounded inF, is equicontinuous onF. Hence, by Alaoglu’s theorem, it is relativelyσ (F,F)-compact.

This, together with theσ (F,F)-closedness implies thatP0isσ (F,F)-compact.

Similarly, as forP0, all setsPn, n∈N, areσ (F,F)-closed, and thereforeσ (F,F)- compact. Any finite intersection

{Pn; 0≤n≤m} =Pmis nonempty. Hence there existsx∈

{Pn; n∈N} ⊂F. To show that

{Pn; n∈N}contains only one element, takey∈F, y≠x. Then there exists aτ-neighborhoodUof zero such thaty∉x+U.

Take aτ-closed, balanced, convexτ-neighborhoodV of zero such thatV−V⊂U.

There existsn∈Nsuch thatxp−xq∈Vfor anyp,q≥n. This implies thatxp∈xn+V forp≥n, Pn⊂xn+V, andxn∈x+V.

Finally,Pn⊂xn+V ⊂x+V+V⊂x+U. But y∉x+U, hencey ∉Pn and y∉ {Pn; n∈N}.

This implies the existence of an upper-triangular matrixΛ=(λnm)with allλnm≥0, only a finite number of nonzero entries in each row, and the sum of all entries in each row is equal to 1, such that the sequence

wn=

∞

m=nλnmxm:n∈N

(3.1)

SEQUENTIAL COMPLETENESS OF INDUCTIVE LIMITS 421 converges toxin the topology ofF. Thenwn→xalso in the weaker topologyτ.

Given a balanced convexτ-neighborhoodUof zero, there existp, q∈Nsuch that wn−x∈Uforn≥pandxm−xn∈Uform≥n≥q. Then forn≥max(p,q), we have

x−xn=

x−wm +

wn−xn

=

x−wm +

∞ m=nλnm

xm−xn

∈U+U, (3.2)

that is,xn→xinE.

We have proved a little more:if eachEnis sequentially complete, then any Cauchy sequence inE=indEnwhich is bounded in someEn, converges to an element inEnin the topology inherited fromE, but not necessarily in the topology ofEn.

In [3], Kuˇcera and McKennon constructed a regular quasi-incomplete inductive limit of Banach spaces and they asked about the existence of a sequentially incomplete regular inductive limit of Banach spaces. Theorem 3.1 provides a negative answer.

We do not know whether sequentially complete inductive limit of sequentially com- plete spaces is regular. Nevertheless, we can at least claim the following.

Theorem3.4. SupposeEnis closed inindEnfor everyn∈NandindEnis sequen- tially complete. ThenindEnisα-regular.

Proof. LetB⊂indEn be balanced, convex, closed and bounded. Then the space EB, with the topology generated by the Minkowski functional of B, is Banach (see Lemma 3.2). PutBn=B

En, Fn=EB

En, n∈N, and equipFn with the topology generated byBn.

SinceEnis closed in indEnandBnis closed inEn, Fnis a Banach subspace ofEB. We haveEB=

{Fn; n∈N}, hence by the Baire’s Category theorem, there exists n∈Nsuch thatFncontains an open set ofEB. This implies thatBnabsorbsBandB is contained inEn, i.e., indEnisα-regular.

References

[1] C. Bosch and J. Kucera,Sequential completeness and regularity of inductive limits of webbed spaces, to appear in Czechoslovak Math. J.

[2] J. Kucera,Sequential Completeness of LF-spaces, to appear in Czechoslovak Math. J.

[3] J. Kucera and K. McKennon,Quasi-incomplete regular LB-space, Internat. J. Math. Math.

Sci.16(1993), no. 4, 675–678. MR 94i:46009. Zbl 815.46005.

Claudia Gómez: Department of Pure and Applied Mathematics, Washington State University, Pullman, WA99164-3113, U SA

E-mail address:[email protected]

Jan Kuˇcera: Department of Pure and Applied Mathematics, Washington State Uni- versity, Pullman, WA99164-3113, U SA

E-mail address:[email protected]