COUNTABLE-CODIMENSIONAL SUBSPACES OF LOCALLY CONVEX SPACES

Transcription

1 COUNTABLE-CODIMENSIONAL SUBSPACES OF LOCALLY CONVEX SPACES by J. H. WEBB (Receved 9th December 1971) A barrel n a locally convex Hausdorff space E[x] s a closed absolutely convex absorbent set. A a-barrel s a barrel whch s expressble as a countable ntersecton of closed absolutely convex neghbourhoods. A space s sad to be barrelled {countably barrelled) f every barrel (a-barrel) s a neghbourhood, and quas-barrelled (countably quas-barrelled) f every bornvorous barrel (ff-barrel) s a neghbourhood. The study of countably barrelled and countably quas-barrelled spaces was ntated by Husan (2). It has recently been shown that a subspace of countable codmenson of a barrelled space s barrelled ((4), (6)), and that a subspace of fnte codmenson of a quas-barrelled space s quas-barrelled (5). It s the object of ths paper to show how these results may be extended to countably barrelled and countably quas-barrelled spaces. It s known that these propertes are not preserved under passage to arbtrary closed subspaces (3). Theorem 6 shows that a subspace of countable codmenson of a countably barrelled space s countably barrelled. Let { } be an expandng sequence of subspaces of E whose unon s E. Then ' s f] E' n, and the reverse ncluson holds as well (1) f ether () E'\o(E', E)~\ s sequentally complete or () E'[f(E', )] s sequentally complete, and every bounded subset of E s contaned n some E n. Theorem 1. Let E[%\ be a locally convex space wth r = n(e, E') (the Mackey CO topology). Suppose E = (J E n, where {E n } s an expandng sequence of subspaces of E. If E' = p) E' n, then E s the strct nductve lmt of the sequence { }. I Proof. Let F[%] be any locally convex space, and T: E^F& lnear mappng whose restrcton T n to E n s contnuous. We show that T s then contnuous, whch proves the result. Let/eF'. Then the composte mappng f T n : E n -+K (scalars) s contnuous,.e. f T n ee' n for each n. Hence/ T e E', so T s a(e, E') - o(f, F') contnuous, hence \(E, E') f(f, F') contnuous, hence T X contnuous. E.M.S. L

2 168 J. H. WEBB The followng result has been proved by M. de Wlde and C. Houet (9). Theorem 2. Let E[x] be a locally convex space, wth E = [j E n, where { } s an expandng sequence of subspaces. Then E s the nductve lmt of {E n } n ether of the followng cases: () E s countably barrelled; () E s countably quas-barrelled, and every bounded subset of E s contaned n some E n. Note that Theorem 1 s not a generalzaton of Theorem 2, for although t s true that the dual of a countably barrelled (countably quas-barrelled) space s weakly (strongly) sequentally complete, there exst countably barrelled spaces E[x] wth T =t= n(e, E'). In fact, Theorem 1 s false f the condton x = ({E, ") s dropped. (Consder E = (j> wth the topology a{4>, co) and E n = {xe 4>: x t = 0 V/>«}.) However, both Theorems 1 and 2 generalze Valdva's result ((6), Corollary 1.5). Theorem 3. Let E be a countably barrelled {countably quas-barrelled) space, and F a closed subspace of E of countable codmenson {of countable codmenson, and such that for every bounded subset B of E, F s of fnte codmenson n span {FKJB}). Then F s countably barrelled {countably quasbarrelled). Proof. Let {x n } be a sequence n E formng a base for a complementary subspace G of F. Put E t = F, E n = span {E n^1, jc n _j} («> 1). Then E = [} E n and by Theorem 2, E s the strct nductve lmt of the sequence {E n }. Snce F s closed, each E n s closed. Consder the projecton map n: E-*F, parallel to G. The restrcton n n : E n -+F s contnuous, snce F s closed and of fnte codmenson n E n. Snce E s the nductve lmt of the sequence {E n }, n s contnuous. It follows that F has a closed complement n E, and that F s somorphc wth a quotent of E by a closed subspace. Snce the property of beng countably barrelled (countably quas-barrelled) s preserved when passng to quotents ((2) Corollary 14), F s countably barrelled (countably quas-barrelled). Corollary. A closed subspace of fnte codmenson of a countably quasbarrelled space s countably quas-barrelled. A smple adaptaton of Theorem 3 enables us to prove a correspondng result for quas-barrelled spaces: Theorem 4. Let E be a quas-barrelled space, and F a closed subspace of E of countable codmenson, such that for every bounded subset B of E, Fs of fnte codmenson n span {FKJB}. Then F s quas-barrelled.

3 COUNTABLE-CODIMENSIONAL SUBSPACES 169 To llustrate the relevance of the condton on the bounded sets mposed n Theorems 3 and 4, we gve the followng examples. Example 1. Let be a countably barrelled space and F a closed subspace of countable codmenson. Let B be a bounded subset of E. Let {F n } be constructed as n Theorem 3. Snce s a strct nductve lmt of closed subspaces, t follows that B s contaned n some F n. So F s of fnte codmenson n span {F\JB}. Example 2. Let E and F be subspaces of the sequence space I 1, defned as follows: E = {xel 1 : x 2n+1 = 0 for all but fntely many «} F= {xel 1 : x 2n+1 = 0 for all n}. Gve the topology of coordnatewse convergence. Then F s a closed subspace of countable codmenson n. If B = {x e Z 1 : x n ^ 1 for all n} then B s a bounded subset of, but F s not of fnte codmenson n span {F\jB}. The man problem s to remove the hypothess that s closed from Theorem 3. The countably barrelled case may be dealt wth by means of a very useful result, due to Saxon and Levn (4). Our proof s a smplfed verson of the orgnal. Theorem 5. Let Ebea locally convex space such that E'[a(E', )] s sequentally complete. If A s a closed absolutely convex subset of E such that span A s of countable codmenson n E, then span A s closed. Proof. Let = span,4 span {* } where {x n } s a lnearly ndependent sequence. We construct g k e ' such that g k (.x t ) = S k and g k (a) = 0 for each ae A. The constructon s as follows: Let B r = T{A,X,..., x k _ 1, x k+l,...,x r } (r>k). Then B r s absolutely convex and closed, and x k $ rb r. By the Hahn-Banach Theorem, there exsts / r e ' such that f r (x k ) = 1 and j/ r (x) g - for each xeb r. The sequence {fr}r>k ' s easly seen to be a(e', )-Cauchy, hence converges to some g k e ' whch s as requred. Snce span A = f) g l k ($>), span A s closed. If span A s of fnte codmenson, the same result holds wth only mnor alteratons n the proof. Theorem 6. If F s a subspace of countable codmenson of a countably barrelled space E, then F s countably barrelled. Proof. Snce F s countably barrelled by Theorem 3, t s suffcent to consder the case when F s dense n. Let U = f] U n be a <r-barrel n F.

4 170 J. H. WEBB Then E7c f] U n = V, and each U n s a neghbourhood n E, snce F s dense. Snce the dual of a countably barrelled space s weakly sequentally complete ((2) Theorem 5), span V s closed, hence U s absorbent. So V s a a-barrel, therefore a neghbourhood n E. Snce Vr\F = U, U s a neghbourhood n F. Valdva ((7), Theorem 4) has proved the followng result: Let E be a sequentally complete 3>3F space. If G s a subspace of E, of nfnte countable codmenson, then G s a QSF space. Snce a sequentally complete 3)3F space s countably barrelled, and the property of havng a fundamental sequence of bounded sets s nherted by all subspaces, Theorem 6 above s an extenson of Valdva's result. We now examne the problem of removng the hypothess that F s closed from the countably quas-barrelled case of Theorem 3. We denote by E + the set of all sequentally contnuous lnear functonals on E (see (8)). Note that whle the elements of " are gven by closed hyperplanes n E, the elements of E + are gven by sequentally closed hyperplanes n E. Lemma 7. Let E be a locally convex space wth E' = E +. Let F be a sequentally closed subspace of E such that for every bounded subset B of E, F s of fnte codmenson n span {FKJB}. Then F s closed. Proof. Let {x a : a e A} be a set of ponts n E lnearly ndependent modulo F, whch, together wth F, span E. For each a e A, defne Then H a s a hyperplane n E. Let {a n } be a sequence n H a convergng to a 0. Then there are ponts x f,..., x Pm (P t e A) such that {a n } <= F+ span {*,,,..., x,j <= H x. Snce F+STpan{x fl,..,*/»,} s sequentally closed, a o eh a, whch shows that H a s sequentally closed. Snce E' = E +, H a s closed. But F = f] H a, so F s closed. Corollary 8. Let E be a locally convex space, wth E' = E +. If F s a subspace of E such that for every bounded closed absolutely convex set B, FnB s closed, and F s of fnte codmenson n span {FnB}, then F s closed. A smlar result s proved by Valdva ((7) Lemma 4) assumng that E s a SlSF space, nstead of " = E +. The above s not a generalzaton of Valdva's result, for there exst 9>& spaces E wth E' =j= E* (see (8)). Theorem 9. Let E be a countably quas-barrelled space wth E' = E +. If F s a subspace of E such that F s of countable codmenson n E, and such that F s of fnte codmenson n span {FKJB} for every bounded set B, then F s countably quas-barrelled. IEX

5 COUNTABLE-CODIMENSIONAL SUBSPACES 171 Proof. Case 1: F closed. See Theorem 5. Case 2: F dense n E. Let U = f] U n be a bornvorous <7-barrel n F. Then Oc f] U n. Let G = span U. We show () U s bornvorous n G, () G = E. () Let 5 be a bounded subset of G. Snce G=>F, there exsts a fntedmensonal subspace M of G such that B<=.F+M = LcG. Now UnL s the closure of U n L, F s of fnte codmenson n L and E7nL s absorbent n L. By a result of Valdva ((7) Lemma 1) UnL s bornvorous n L, so U absorbs B. () Snce G=>F, and F s dense, t s suffcent to prove that G s closed. Let B be a bounded absolutely convex closed subset of E. Then for some a, GnBcctXJ, so GnB s closed. By Corollary 8, G s closed. Thus f) U n s a bornvorous a-barrel n JE 1, hence a neghbourhood. Therefore U = ( P) U w I nf s a neghbourhood n F. \ / Case 3: Arbtrary F. Ths follows at once from Cases 1 and 2. Ths result s a varaton of one of Valdva, who proves ((7) Theorem 2) that a subspace Fof a 2& space E, such that Fs of fnte codmenson n span {FuB} for every bounded set B, s tself a 9)SF space. Such a subspace s necessarly of countable codmenson, whle our result above requres only that the closure of the subspace concerned be of countable codmenson. Corollary. If E s a countably quas-barrelled space wth E' = E +, and F s a subspace of fnte codmenson n E, then F s countably quas-barrelled. It s not known whether the condton " E' = E + " can be omtted from the Corollary. An easy adaptaton of the proof of Theorem 9 yelds the followng result: Theorem 10. Let E be a quas-barrelled space wth E' = E +. If F s a subspace of E such that F s of countable codmenson n E, and such that F s of fnte codmenson n span {F<oB} for every bounded set B, then F s quasbarrelled. Corollary 11. Let E be a bornologcal space. If F s a subspace of E such that F s of countable codmenson n E, and such that F s of fnte codmenson n span {FuB} for every bounded set B, then F s quas-barrelled. Proof. A bornologcal space Es quas-barrelled and satsfes: E' = E* (8). The author s ndebted to the referee for several helpful suggestons.

6 172 J. H. WEBB REFERENCES (1) M. DE WILDE, Quelques thdor^mes d'extenson de fonctonnelles lneares, Bull. Soc. Roy. Sc. Lege 35 (1966), (2) T. HUSAIN, Two new classes of locally convex spaces, Math. Ann. 166 (1966), (3) S. O. IYAHEN, Some remarks on countably barrelled and countably quasbarrelled spaces, Proc. Ednburgh Math. Soc (1967), (4) S. SAXON and M. LEVIN, Every countable-codmensonal subspace of a barrelled space s barrelled, Proc. Amer. Math. Soc. 29 (1971), (5) M. VALDIVIA, A heredtary property n locally convex spaces, Ann. Inst. Fourer {Grenoble) 21 (1971), 1-2. (6) M. VALDIVIA, Absolutely convex sets n barrelled spaces, Ann. Inst. Fourer (Grenoble) 21 (1971), (7) M. VALDIVIA, On 3>& spaces, Math. Am. 191 (1971), (8) J. H. WEBB, Sequental convergence n locally convex spaces, Proc. Cambrdge Phlos. Soc. 64 (1968), (9) M. DE WILDE and C. HOUET, On ncreasng sequences of absolutely convex sets n locally convex spaces, Math. Ann. 192 (1971), UNIVERSITY OF CAPE TOWN SOUTH AFRICA