PII. S0161171203205202 http://ijmms.hindawi.com © Hindawi Publishing Corp.
ON SEQUENTIALLY RETRACTIVE INDUCTIVE LIMITS
ARMANDO GARCÍA
Received 27 May 2002
Every locally complete inductive limit of sequentially complete locally convex spa-ces, which satisfies Retakh’s condition(M)is regular, sequentially complete and sequentially retractive. A quasiconverse for this theorem and a criterion for se-quential retractivity of inductive limits of webbed spaces are given.
2000 Mathematics Subject Classification: 46A13, 46A30.
1. Introduction. Throughout the paper,{(En,τn)}nis an inductive sequence
of locally convex spaces and(E,τ)=ind(En,τn)is its inductive limit. Recall
that E is regular if every bounded subset inE is contained and bounded in one of the steps, andEis sequentially retractive if every null sequence inE converges to zero in some step. We say thatEsatisfies the Retakh’s condition (M)if in every spaceEn, there is an absolutely convex neighborhood of zero
Unsuch that
(1) Un⊂Un+1for everyn∈N;
(2) for everyn∈N, there ism > nsuch that all the topologies of the locally convex spacesEk, for k≥m, coincide onUn. Equivalently,τ and τm
coincide onUn.
We assume that every suchUnisτn-closed and thatτn+1andτinduce the
same topology onUn, which we do without loss of generality.
Finally, we say thatE satisfies condition(Q)(see [8]) if part (1) in (M)is dropped.
Vogt in [7] studied condition(M)for LF-spaces, that is, for inductive limits of metrizable and complete (equivalently and sequentially complete) locally convex spaces. He obtained several important results about them; for exam-ple, that on LF-spaces, condition (M)implies completeness, regularity, and sequential retractivity. Recently, Wengenroth in [8] proved the following very important result on LF-spaces: condition (M), condition(Q), acyclicity and sequential retractivity are equivalent.
On the other hand, Gómez-Wulschner and Kuˇcera in [2,3] studied sequen-tial completeness and weak regularity conditions for inductive limits of se-quentially complete spaces. They have shown that a regular inductive limit of sequentially complete spaces is sequentially complete [3].
The last part is devoted to webbed spaces (definitions are recalled in that section). We present a quasiconverse to Theorem 2.5and a criterion for se-quential retractivity.
2. Regularity and sequential retractivity. Recall that a diskDin a locally convex spaceFis an absolutely convex, bounded and closed subset. We write (FD,ρD) to denote a normed space, whereFD=spanDand ρD is the norm
topology generated onFDby the Minkowski’s functional ofD; equivalently,ρD
is generated by the basis of neighborhoods{λD:λ >0}. Note that closedness is not necessary for the Minkowski’s functional to be a norm.
In order to obtain the first theorem, we need a technical lemma and a pair of useful propositions.
Lemma 2.1. If E=indEn satisfies condition(M) for the sequence(Un)n,
thenUjE=∞k=jUjEkfor everyj∈N.
Proof. Sinceτ restricted toEkis coarser thanτk, we haveUjEk⊂UjEfor
k≥j. So,∞k=jUjEk⊂UjE. Conversely, letx∈UjE. There exists a net(xα)α⊂
Uj such that xα →τ x. This implies that there existsn≥j and λ >0 such
that λ{(xα)α,x} ⊂Un. Since τ and τn+1 coincide onUn, λxα τn+→1 λx; so,
xα τn+→1 x. Hence,x∈UjEn+1.
The next proposition is the key toTheorem 2.5.
Proposition2.2. Let every(En,τn)be locally complete. IfE=indEn
satis-fies condition(M), then every Banach diskB⊂E is contained and bounded in someEn.
Proof. LetB ⊂E be a Banach disk. By [5, Proposition 8.5.20], there
ex-istsp∈Nsuch thatB⊂pUpE=p∞k=pUpEk, the last identity follows from Lemma 2.1.
SinceB is τ-closed andτ-bounded,B∩Ek isτk-closed andB∩pUpEk⊂B
is τ-bounded, for every k≥ p. Let Bk = B∩pUpEk. We assume that every
Uk is τk-closed, then(1/p)Bk⊂UpEk ⊂UkEk =Uk for everyk≥p. By
con-dition (M), τ and τk+1 coincide onUk, then (1/p)Bk is τk+1-bounded. Now,
the local completeness ofEk+1implies thatBkEk+1is a Banach disk inEk+1, so
(EB
kEk+1,ρBkEk+1)is a Banach space continuously embedded in(Ek+1,τk+1). Note that for everyk≥p,
BkEk+1=B∩pUpEk Ek+1
⊂B∩pUpEk+1 Ek+2
=Bk+1Ek+2. (2.1)
This implies thatBkEk+1is contained inBk+1Ek+2∩EB
kEk+1; therefore(EBkEk+1, ρB
It follows that ind(EB
kEk+1,ρBkEk+1)is an LB-space. In order to finish the proof, we prove that this is a nonproper LB-space. In other words, we show that there existsk0∈Nsuch that(EB
k0Ek0+ 1,ρB
k0Ek0+
1)=(EB,ρB).
SinceBisτ-closed andBk⊂B,we haveBkEk+1⊂B.AndBkEk+1⊂B∩EBkEk+1 which implies that the identity mapik:(EBkEk+1,ρBkEk+1)→(EB,ρB)is
continu-ous for everyk≥p. On the other hand,
B=B∩p
∞
k=p
UpEk= ∞
k=p
B∩pUpEk= ∞
k=p
Bk⊂ ∞
k=p
BkEk+1⊂B. (2.2)
This means that span(B)=∞k=pspan(BkEk+1). Therefore, the identity map
i: indEB
kEk+1,ρBkEk+1
→EB,ρB (2.3)
is continuous and onto. By the open mapping theorem (see [5, Theorem 8.4.11]), the inverse identity map
j:EB,ρB →ind
EB
kEk+1,ρBkEk+1
(2.4)
is continuous. By Jarchow [4, Corollary 5.6.4], the space(EB,ρB)is continuously
embedded in some(EB k0Ek0+1
,ρB k0Ek0+1
).
We conclude thatBis contained and bounded in(Ek0+1,τk0+1).
Corollary2.3. Let every(En,τn)be locally complete. If E=indEn is
lo-cally complete and satisfies condition(M), thenEis regular.
Proposition2.4. Let every(En,τn)be sequentially complete. If E=indEn
is regular and satisfies condition(M)for a sequence(Un)n, thenEis sequentially complete and sequentially retractive.
Proof. Let(xl)lbe a Cauchy sequence in(E,τ). Then,A= {xl:l∈N}is a
τ-bounded set. So, there existsn∈N, such thatA⊂EnandAisτn-bounded.
There existss >0 such thatsA⊂Un. Sinceτandτn+1coincide onUn, it follows
that(sxl)lisτn+1-Cauchy, thenτn+1-convergent tosx0, for somex0∈En+1,
hence(xl)lis convergent tox0 in(E,τ).
In an analogous way, it is straightforward to show that(E,τ)is sequentially retractive.
From the preceding results we conclude the following theorem.
Theorem2.5. Let every(En,τn)be sequentially complete. IfE=indEn is
3. Sequential retractivity on webbed spaces. We give now two results on sequential retractivity for certain webbed spaces. For convenience, we recall some basic facts about webs which we need. For more information about the basic properties of webs, we refer the reader to the works of De Wilde [9], Jarchow [4], and Robertson [6].
A strand of a webWon a locally convex space(F,τ)is a collection of mem-bers ofW, one from each layer, with the(k+1)th member of the strand con-tained in thekth member. Strands will be denoted by (Wk)k. A web on F is
compatible withτ if for each neighborhood of zeroUin(F,τ)and for each strand(Wk)kofW, there isk0such thatWk0⊂U.
Following the idea of Wengenroth in the proof of [8, Proposition 2.3], we get a quasiconverse forTheorem 2.5. To simplify the notation in this proposition, we useWk∈Wto denote Wk1,k2,...,kr ∈W, that is, write only one index as for the elements of a specific strand.
Proposition3.1. Let every(En,τn)be a webbed space andE=indEn
se-quentially retractive. Then, for everyN∈Nthere isn≥Nand an elementWkN of the webW(N)onE
N, for somek=k(N), such thatτandτncoincide onWkN.
Proof. Suppose that this proposition is not true. So, there existsEn
0such that for every element of its web, sayWn0
k ∈Wn0, and for eachN∈Nthere
existsn > Nsuch thatτnrestricted toWkn0is strictly coarser thanτNrestricted
toWn0
k .
For suchn0, fix an element of the webWkn00∈Wn0. LetN=n0, then there
exists n1> n0 such that τn1 restricted toW
n0
k0 is strictly coarser thanτn0. So, there is a sequence(xk0
l )l⊂Wkn00, which isτn1-null but notτn0-null. Find an element of the(k0+1)th layer of the webWkn00+1∈Wn0, such thatW
n0
k0+1+ Wn0
k0+1⊂W
n0
k0, andn2> n1such thatτn2 restricted toW
n0
k0+1is strictly coarser thanτn1restricted toW
n0
k0+1. Then, there is a sequence(x
k0+1
l )l⊂Wkn00+1, which is τn2-null but notτn1-null. In this way, determine a strand(W
n0
k0+k)k of the webWn0, an increasing sequence of natural numbers(nk)k, and a collection of sequences[(xk0+k
l )l]ksuch that every(xlk0+k)l⊂Wkn00+kisτnk+1-null but not τnk-null.
LetUbe a neighborhood of zero in(E,τ). Then,U∩En0is a neighborhood of zero in(En0,τn0), so there existsK∈Nsuch thatWkn00+k⊂U∩En0⊂Uifk > K.
It implies that(xk0+k
l )l⊂Ufor everyl∈Nifk > K. Now, ifk≤K, then(xlk0+k)l
is aτ-null sequence, since it is aτnk+1-null sequence. Hence, arranging the double-indexed sequence in any way into a single indexed sequence, it results aτ-null sequence. So, this sequence should be convergent in someEm since
Eis sequentially retractive. But this is not possible, since the sequence is not convergent in anyEm. Hence the proposition is true.
if inProposition 3.1everyone of the corresponding elements from the webs, where the topologies coincide isτn-neighborhoods of zero, thenE satisfies
condition(Q), and hence by [8, Proposition 2.5],Esatisfies condition(M). Following Gilsdorf [1], a locally convex spaceF is sequentially webbed if it has a compatible webW such that for every null sequence(xm)m in(F,τ),
there exist a strand(Wk)kand a natural numberMkfor everyk∈N, such that
xm∈Wkfor allm≥Mk. To simplify, we denote this condition by(#).
From [4, Corollary 5.3.3(b)], the inductive limitE=indEn of a numerable
sequence of webbed spaces is again webbed and admits a completing webW such thatW(n)=Enfor everyn∈N. Moreover, thekth layer of the webWon
E is the collection of members of thekth layer in the spacesEn. In the next
theorem, we use such a web onE=indEnin order to characterize sequential
retractivity for inductive limits of sequentially webbed spaces.
Theorem3.2. Let every(En,τn)be a sequentially webbed space.E=indEn
is sequentially retractive if and only ifEis sequentially webbed.
Proof. Suppose that E is sequentially retractive. For any null sequence
(xm)m in (E,τ), there exists n∈N such that (xm)m is a null sequence in
(En,τn). So, there is a strand(Wk(n))kof the webW(n) onEnsatisfying(#)on
En. Note that by the form of the web onE,(Wk(n))kis also a strand for the web
WonE. So,Eis sequentially webbed. Conversely, let(xm)mbe a null sequence
in(E,τ); since E is sequentially webbed, there is a strand(Wk)k of WonE
satisfying(#). By the form of the web onE,W1=W(n)=Enfor somen∈N.
So, this strand is contained inEnand it is a strand ofW(n) onEn. Now, since
W(n)is compatible withτn, for everyU-neighborhood of zero in(En,τn), there
existsk∈Nsuch thatWk⊂U. Hence,xm∈Ufor allm≥Mk.
Acknowledgments. The author is very grateful to the “Centre de Recerca
Matemática,” Barcelona, Spain for the kind hospitality during the preparation of part of this paper, and also to the referee for many valuable comments.
References
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Armando García: Instituto de Matemáticas, UNAM, Zona de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, México, DF 04510, Mexico
