Math. Z. t78, 263-269 (I981) Mathematische Zeitschrift

�9 Springer-Vertag t98t

On Totally Barreled Spaces

Manuel Valdivia 1 and Pedro P~rez Carreras z

i Facuttad de Matem~ticas, Dr. Motiner s.n., Bmjasot.Yalenc~a, Spain z E.S.I. Industriates, C~'era s.n., Valencia, Spain

The linear spaces we shall use are defined over the field K of the real or comNex numbers. If A is a subset of a linear space we denote by (A) and [A] its convex hull and linear hull respectively. A space E is a Hausdorff locally convex topological linear space and its topological dual is denoted by E'. If B is a bounded absolutely convex subset of a space E we denote by Es the normed space over the linear hull of B. Given spaces E and F we denote by E | the tensor product E | endowed with the projective topology. A space E is Baire-tike, [6], if given an increasing sequence (A,) of absolutely convex closed sets of E covering E there is an integer n o such that A,o is a neigh- bourhood of the origin. A space E is suprabarreted, [9], if given an increasing sequence (E,) of subspaces of E covering E there is a natural nun:ber n o such that E,o is barreled and dense in E. Every suprabarreled space is a Baire-like space and an (LF)-space which is metrizabte and non-complete is an example of Bairelike space, [1], which is not a suprabarrelted space by an obvious application of Pt~tk's homomorphism theorem, [4]. We have the following result:

Theorem I. Let E be a Banach space and let F be a barreled dense subspace of E which is not suprabarreted. 77~en there is a proper subspace G of E containing F which is an (LF)-space for a topology T finer than the topology induced by E.

Proof t. Construction of G. We take an increasing sequence (F~,) of subspaces of F covering F and such that F,~ is not barreled, n= 1, 2, .... There is in F;, a bounded barrel T,, which is not a neighbourhood of the or~in in .~. Let U;, be the closure of "~ in E and let Pf be the set Uvc',L~+lc~...c',U,, with p,n =1,2, . . . and n>p. If Gp denotes the subset of E of all those etements which are absorbed by t~ p for all n, it follows that @ is a linear subspace of E containing F v and such that G v : Gp + a. We suppose that every Gp as well as G

= U Gp is endowed with the topology induced by E. Let Tp be the topology

on Gp such that G~[~] is the projective limit of the sequence (Ev~: n=p,p +1 .... ) of Banach spaces. Clearly Gv[Tp] is a Fr6chet space, T~, is finer than T~+~ on Gp and 7) is finer than the topology of E induced on G e. We provide G with the inductive topology T of the sequence of Fr6chet spaces (G~,[7~,]: p =: ,2 , ,-8-

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264 M. Valdivia and P. P6rez Carreras

2. G is distinct from E. Since every l/~ is a bounded set of E it follows that the canonical injection I: G [ T ] ~ G is continuous and therefore 1-1: G ~ G[T] has dosed graph in G x G[T]. If G coincides with E we apply Grothendieck's dosed graph and factorization theorems in order to find a natural number q such that E is continuously imbedded in Gq and thus I/; q = Uq is neighbourhood of the origin in Fq, and we arrive to a contradiction, q.e.d.

Theorem2. Let E be an (LF)-space which is metrizable and non complete an let F be a dense subspace of E which is barreled. Then, F is not suprabarreled.

Proof. There is an increasing sequence (En) of subspaces of E covering E such that E,~=E~+I, n = l , 2 , ..,, and such that E, is not barreled, n = l , 2 .. . . , accord- ing to Ptak's homomorphism theorem. Setting F,=E,c~F we have an increas- ing sequence (F,,) of subspaces of F covering F. We apply Amemiya-Komura's theorem, I l l , to obtain a positive integer n o such that F,, n>__no, is dense in F and thus in E. If F is suprabarreled there is a positive integer p > n o such that Fp is barreled and therefore Ep is barreled since Fp=E. That is a contradiction, q.e.d.

Definition 1. A space E is totally barreled if given a sequence (E,) of subspaces of Ecoc, ering E there is an integer n o such that E,o is barreled and its closure in E is of finite codimension in E.

We recall a definition, [7J: A space E is unordered-Baire-like if given a sequence A, covering E of absolutely convex closed subsets of E there is a positive integer n o such that A,o is a neighbourhood of the origin, It is obvious that every unordered-Baire-like space is totally barreled and that every totally barreled space is suprabarreled.

Example 1. In [-3] it is shown that the subspace E of #1 of all those sequences which are null save in an index subset of N of zero density, is a barreled space which is Baire-like, since it is metrizable. We shall show that E is even totally barreled. (In [6] it is observed that E is not unordered-Baire-like.) It is enough to show that every time E is the union of a sequence of subspaces there is one of them which is barreled. Indeed, let us suppose the former assertion true and suppose E is the union of a sequence (H,) of subspaces of E. If we consider the sequence (H1,H 2 . . . . , l~lp .. . . ), of closures taken in E, we claim the existence of at least one subspace ]-/p0 such that/~po is of finite codimension in E: Suppose this is not true; then for every p we select an infinity of linearly independent

P P P vectors (xl,x2 . . . . ,xh, . . . ) which do not belong to /tp and we call Gp the set

[Hpw(xf,x~, ...,Xh,P ...)3. Since E--- U Gp there is an integer P0 such that Gpo p = l

is barreled, by our assumption. But then Gpo is the inductive limit of the sequence - - PO Po ([Hpo W(Xl ,x2 , ' " , x,P~ 1,2 .... by [8] and thus Gpo contains qo which is a con-

tradiction, since E is metrizable. Therefore if we call f f l the family of subspaces of (H,) such that its closure in E is of finite codimension in E and i f 2 : = (H,)\,~'~ it follows that ~ has to cover E, since if2 does not cover E ([7], 4.1 Theorem). We apply again our assumption and we are done. Let us show that if E is the union of a sequence (L,) of subspaces of E there is an integer n o such that L,o is barreled. We write e,, n = l , 2 . . . . , for the unit vectors of ~,1 and A for the

On TotaiIy Barreled Spaces 265

set (el, e2 . . . . , ep,...). Suppose [L, u A ] , n - l , 2 . . . . ,E, is not which is not a neighbourhood Clearly ~, does not absorb A closed absolutely convex hull sequence of natural numbers =1,2, . . , Let

l < q ~ < . . . < q ~ < ,,,

be a sequence of natural numbers of zero density. Select a natural (1,p~)_> 1. We write p~ =p~. By recurrence suppose we have constructed

none of the subspaces (L,) is barreled. If E,, is barreled, [5] and [8]. Let 7;, be a barrel in E, of the origin in E, and let l~; be its closure in E. for any n since the dosed unit ball of E is the of A. Thus we select tbr every n an increasing (n,I), (n,2),..., (n,s),.., such that e(,,:)(~sV,,, n,s

(1)

I 2 n (t,p~),(2,p~),(1,-p~) . . . . ,(n,p~),

( n - 1,p~) . . . . ,0 ,p~)

such that a finite subsequence of (1), t < r I < h < . . . < r e , has been selected satisfying

1 <(1,pl)__< q < (2,p2)_< r2 < (1,p~)< ... <r,<(n,p"l)

=<t~+ i < ... <(1,p~)_~ r/>

We take an element ( n+ l , p ] +a) in the sequence ( n + l , 1), (n+l ,2 ) . . . . , (n + 1,p) . . . . such that (n + 1, p]+ ~)>~? and we select in (1) the first element r~, such that l ~ ( n + l , p ] + l ) . In the sequence (n,1), (n, 2) .. . . ,(n,p) ... . we select an ele- ment ~ + 1 (n, pa )>r~ and we continue in that way until we select (1,p~+l). Let F be the Banach space of all those sequences of E such that are null except in the positions indexed by (2). Since F = () (m l/;, a F) and since F is Baire there is

an integer n o such that I/~o:~F is a neighbourhood of the origin in F and therefore V,, ~ c~ F absorbs the bounded set

{e(,,,.p~o), %o,p:~.+~) . . . . . %o,p~5+:), ... },

~ : ~ V~o, j=0 ,1 ,2 , q.e.d. contradicting the relations e(,~o.pT~+/~ ,_vj+~ ...

The proof of the following theorem is easy and will be omitted.

Theorem 3. a) I f F is a dense subspace of a space E and ~' F is totally barrelled, then E is totally barreled.

b) I f E is a totally barreled space and if F is a closed subspace of E then E/F is a totally barreled space.

Theorem 4. Let E be a totally barreled space and let (E,) be a sequence of subspaces of E covering E. There exists a positive integer p such that Ep is totally barreled and such that its closure Ev in E is of finite codimension.

Proof According to [7], 4.1 Theorem it is easy to see that a subsequence (F,,) of (E,) can be selected such that ~ , n = t, 2 . . . . . is of finite codimension in E. If F~ is not totally barreled, n = 1, 2,. . . , there is a sequence (F,,)v_ ~ 2 of subspaces of ~, covering F~ and such that F, is not barreled or it is 'barreled but the

�9 l e

codimension of its closure in F, is mtinite. Let d , be the family of all those

266 M. Valdivia and P. P~rez Carreras

elements of (~p)p=l,2 .... which are not barreled and set N,=(~p)p= 1,2,...\ffn. Setting d = ~ d , and N = ~) N, it follows that ,z /wN covers E; thus, one

n=l n= l element of ~4 w N is barreled and its closure in E is of finite codimension in E. But that can not be true, since if G e d then G is not barreled and if G s N there is an integer n such that G c N,. If (~ denotes the closure of G in F, and

its closure in E, G is of infinite codimension in E, since G--Gc~F, and 0 is of infinite codimension in F,. q,e.d,

According to the last theorem the following can be easily shown to be true,

Theorem 5. Let E be a totally barreled space and let (En) be a sequence of subspaces of E covering E. 7?wre is a subsequence (F,) of (En) which are totally barreled spaces and such that their closures in E are of finite codimension in E.

Theorem 6. Let E be a totally barreled space and let F be a countabIy codimen- sional subspace of E. Then F is totally barreled.

Proof. Let G be a countably dimensional algebraic complement of F in E. and let (F,~) be a sequence of subspaces of F covering F. Setting E , = F , + G we conclude from Theorem 4 the existence of a positive integer p such tha tEp = Fp + G is totally barreled (and therefore a Baire-like space) and such that Ep is of finite codimension in E. If ~ denotes the closure of ~ in F and ~ its closure in E it is clear that ~+G--_Hp is a Baire-like space, since Ep is contained and dense in H v. B_ut since Fp is of countable codimension in Hp and since i~ is closed in H v, Fp is of finite codimension in gp and therefore H v is closed in E; thus / /p coincides with Ep. Then .~ is of finite codimension in E and therefore ~=Fvc~F is of finite codimension in F. According to [8] and [5], Fp is barreled, since it is a countably codimensional subspace of Ep. q.e.d.

Proposition 1. Let E be a barreled space which is not totally barreled. There exists a sequence (Vv;) of closed absolutely convex sets of E covering E such that

W~ n [ ,,~, is of infinite codimension in E, n = 1, 2,...

Proof, Since E is not totally barreled there are two countable families sg and N with d u b covering E (,~r or N can be void) such that if G e d , G is not barreled and if H e N then /7 is of infinite codimension in E. If G e a / l e t T(G) be a barrel in G which is not a neighbourhood of the origin and let U(G) be its closure in E: Clearly, [U(G)] is of infinite codimension in E since otherwise [U(G)] would be barreled as a finite codimensional subspace of a barreled space E, [2], and thus U(G)c~G=T(G) would be a neighbourhood of the origin in G. We are done by taking (W,) as the family {/7: H ~ } w { m U ( G ) : Gssg , r e = l , 2 . . . . }. q.e.d.

For the next proposition we consider a family {Ei: i EI} of spaces and we set E = I ] { E i : ie I} . Let W~, be a closed absolutely convex set of E such that [W,] is of infinite codimension in E and let ~,~ be the family {[W,]: n= 1,2 . . . . } covering E. For every i~ I let ~ be the family { F e ~ : Fc~E i is of infinite codimension in El}. If M is a subset of I we identify I~{EI: i e M } as a subspace of I ] {Ei: ie I} .

On Totally Barreted Spaces 267

Proposition Z Suppose that jbr every F E ~ there is a finite set d (F)~= 1 such that F contains I] {E5 i~I/d(F)}. Then ~here is an index j ~ I such that thefiTmily covers Ej.

ProQf First we claim that ~ = U {~'~: i ~ I}: indeed, if G ~ ~ there is a finite set J = I such that G=H{E~: ieI\,,J} and thus Gc~[I(Ei: i6J) if of infinite codimension in I-I (Ev i ~J); therefore there is an index r ~J such that G c~E~ is of infinite codimension in E~, that is G ~ ~.~ If the desired conclusion were not true then for every index i e l there would be a vector x~E~ such that x i~ U (H: H~o~). Let Li be the linear hull of {x~} and let L=I~[{L~: ieI}, which is a Baire space covered by ~ .

There is a positive integer s such that [W~] is of second category in L and, since -F~] is absolutely convex, [W~] contains L. This is a contradiction, since [Ws]~ ~ for some index i~ t and x ~ r by construction, q.e.&

Proposition 3 ([7], 4.6 Lemma)~ Let (E,) be a sequence of spaces and let ~ be a countable family of closed absolutely convex subsets of E - - I ] {E,: n = 1, 2,...}. I f N covers E and N I = { B e ~ : ~ { E , : n = p , p + l .... }c_B for some p} then = {[B]: B~/;~} covers E.

Proposition 4. Given a J~mity {E~: i~I} of spaces let ~ be a countable family of closed absolutely convex sets of E = ~ {E~: i ~ I}. I f N covers E and N1 = {B ~ N: B~[-[{Ei: i~t \J(B)} , J(B) being a finite index subset of I, then ~ = { [ B ] : B ~ N1} covers E.

Proof We claim that S~31 is non void. Suppose N~ is a void family and set N ={B,: n = l , 2 .... }. Since B, is convex and closed, if it contains E~ for every is.t, B,, contains E. Thus, by assumption, there is an index i,,~ 61 such that E~,,~ ~B, , n=l ,2 , .... Since B,~ does not contain I~{E~: i~l\{i,~}}, there is an index i,, 2 e t such that E~.~r and so on; we determine a countable set of indexes I,

< I such that B~z~E~, ie I , , n = l , 2 . . . . . Let M be I~ I, and set F=H{E~: n = l

i e M}. According to Proposition 3 there is a finite subset J c M and a positive integer q such that Bq~t~ {E,: i ~ M \ J } . But this is a contradiction, sinc~e if we take an integer r such that iar~J , then E cB~ and Eiq,.~B q.

If ~ does not cover E the family 0~ {[B]: B ~ Nt} covers E. But then N~ ={roB: [ B i e r , r e= l ,2 .... } covers E and according to what has been said before, there is a positive integer n and a subset B q ~ a such that n B ~ I ~ {E~: i ~ I \ P } being P a finite subset of I and thus B ~ N , which ~s a contradiction, q.e.d.

Theorem 7. 7he product E--I'I{E~: i~I} of totally barreled spaces is totally barreled.

Proof. We suppose that E is not totally barreled. Since E is barreled we apply Proposition 1 to obtain a countable family {g~: n = t , 2 .... } of dosed ab- solutely convex subsets of E covering E such that [W,] is of infinite codimen~ sion in E, n = l , 2 . . . . . We set ~ = {[I4~]: n=l ,2 , . . .} and according to Proposi~ tion 4 we obtain a subfamily of ~, which we denote by ,N again, such that I4'~

l~ {E~: i~ I\J~}; J~ denoting a finite subset of 1. By Proposition 2 there is an

268 M. Valdivia and P. P&ez Carreras

index j e I such that ~,~ covers E i. where ~ = { F e o ~ - : Fc~Ej is of infinite codimension in Ej}. Since Ej is a totally barreled space there is an index s such that [W, l e ~ , E2c~[W~] is barreled and its closure in Ej is of finite codimension in E2. But this a contradiction since Ejc~[W~] is closed in E2: indeed lvV~c~Ej is a neighbourhood of the origin in Ejc~[W~l and W~nE 2 is closed in Ej. q.e.d.

Theorem 8. Let E be a space such that there is a sequence (H,) of closed hyperpIanes of E covering E and let F be an infinite dimensional space. Then, E | is not totally barreled,

Proof Let L~ be a topological complement of/-/n in E. It follows that

E | = (//~ x L~)|174174174 x F.

Setting G~=I-I~| we claim that (G,) covers E| if z = ~ x ~ | 1 7 4 we i = 1

denote by L the linear hull of {xl,x2, ...,x~}. The sequence ( H ~ L ) covers L and, since L is Baire, there is a positive integer p such that Hp2 L and thus z e @. Since G~ is a closed subspace of E | and of infinite codimension in E | F, we have that E | is not totally barreled, q.e.d.

Corollary 8.1. Let E be a totally barreled space which is not an unordered-Baire- like space and let F be a space. Then E| is totally barreled if and only if F is finite dimensional.

Proof If F is finite dimensional then E| is isomorphic to E aimF which is obviously totally barreled. We suppose now that F is an infinite dimensional space. We can select a sequence (E,) of subspaces of E covering E such that E, is barrelled and E, is of finite codimension in E, n = 1, 2 . . . . Since E is not unordered-Baire-like we have that every E, is of non-zero finite codimension on E. Let H~ be a closed hyperplane of E containing E,. We apply Theorem 8 to obtain that E | is not totally barreled, q.e,d.

Example 2. ~ is the subspace of doo of all those bounded sequences which take at most a finite number of different values. In [11-t it is shown that d~ is suprabarreled. The following result is contained in [10]: ~ H be a metriz- able suprabarreled space and let F be an unordered-Baire-like space. The H | is suprabarreled". Then d~ ~ | 2 is suprabarreled but not totally barreled, according to Theorem 8.

Let E be the totally barreled space which is not unordered-Baire-like considered in Example 1. According to the quoted result contained in [10], E | 2 is suprabarreled but not totally barreled, by virtue of Corollary 8.1.

Open Problem. We do not know if d~ is totally barreled.

References

1. Amemiya, L, Komura, Y.: 0ber nicht vollst~tndige Montelr~iume~ Math. Ann. 177, 273-277 (1968)

On Totally Barreled Spaces 269

2. Dieudonne, J.: Sur les propri6t6s de permanence of certains espaces vectoriels topologiques. Ann. Polon. Math. 25, 50-55 (1952)

3. K/Sthe, G.: Topologische Lineare R~iume I. Berlin-Heidelberg-New York: Springer 1966 4. Ptak, V.: Completeness and the open mapping theorem. Bull. Soc. Math. France 86, 41-74

(1958) 5. Saxon, S., Levin, M.: Every countable-codimensional subspace of a barreled space is barreled.

Proc. Amer. Math. Soc. 29, 91-96 (1971) 6. Saxon, S.: Nuclear and product spaces, Baire-like spaces and the strongest locally convex

topology. Math. Ann. 197, 87-106 (1972) 7. Todd, A., Saxon, S.: A property of locally convex Baire spaces. Math. Ann. 206, 23-32 (1973) 8. Valdivia, M.: Absolutely convex sets in barreled spaces. Ann. Inst. Fourier (Grenoble) 21, (2)

3-13 (1971) 9. Valdivia, M.: On suprabarreled spaces. Lecture Notes in Mathematics (Editor: S. Machado).

Berlin-Heidelberg-New York: Springer (to appear) 10. Valdivia, M.: A class of locally convex Suslin spaces without Cg-web. Preprint 11. Valdivia, M.: On certain normed barreled spaces. Ann. Inst. Fourier 29, 2 (Grenoble) 39-56

(1979)

Received December 11, 1980