Math. AnrL 201,145--148 (1973) (~) by Springer-Verlag 1973

A Note on Locally Convex Spaces*

Manuel Valdivia

In this paper we use vector spaces over the field K of real or complex numbers. We shall use the word "space" meaning "separated locally convex space". Given a space E we denote, as usual, by E' its topological dual and by tr(E', E), ~(E', E) and #(E', E) the weak, strong and Mackey topologies on E', respectively. If d is the family of all the absolutely convex, closed and bounded sets of E'[fl(E', E)] we denote by gb* the topology on E of the uniform convergence on every set of d . 2(E', E) is the topology on E' of the uniform convergence on precompact subsets of E. If A C E, cr(A) denotes the topology on E' of the uniform convergence on finite subsets of A.

Let ~t be an infinite cardinal number. Given a space E we consider the family ~ , of the dosed absolutely convex sets of E'[tr(E', E)], so that B ~ ~ , if and only if every subset of B, whose cardinal number is not larger than ~, is equicontinuous. We denote by g~') the topology on E of the uniform convergence on every element of ~ . In particular ~(~') is the locally convex topology on E, which has as fundamental system of neighbourhoods of the origin the polar sets of the closed absolutely convex sets of E'[a(E', E)], whose infinite countable subsets are equicontinuous. We say that a space E is ~-barrelled, (~-quasi-barrelled), if every bounded subset of E'[a(E', E)], (of E'[fl(E', E)]), whose cardinal number is not larger than ct, is equicontinuous. In particular a space E is o~-barrelled, (og-quasi-barrelled), if every infinite countable bounded subset of E'[a(E', E)], (of E'[~(E', E)]), is equicontinuous. We say that a set B of a space E is ~-separable if it admits a dense subset of cardinal number non-larger than ~.

Lemma. Let A be a set of a space E[o'vJ. I f A is precompact then 8 and ~o,~ coincide on A.

Proof. Let B e ~,o. Since every countable subset orB is equicontinuous it results that B is precompact in E'[2(E', E)], (t-1], 3, p. 74). By this B is precompaet for the semi-norm supl(x,x ') l and, therefore, A is

precompact for the semi-norm supl(x,x ') l , ([1], p. 200), hence A is x I ~ B

* Supported in part by the "Patronato para el Fomento de la Investigaci6n en la Universidad".

1,16 M. Valdivia:

precompact in E[drt~')]. Since E[~'~) 1 has a fundamental system of neighbourhoods of the origin closed in E[81 it results that dr and dr(,o) coincide on A, ([51, (2), p. 385). Q.E.D.

Theorem 1. Let ct be an infinite cardinal number. Let A be a linear set of a space E[drl. I f there exists in A a family {Ai: iE1} of precompact sets, whose union is total in A, so that the cardinal number of I is not larger than ct, then dr and dr~') coincide on A.

Proof Let B e ~ , C ~ o , . Since B is precompact in E'E,t(E',E)] it results that B is separable for the semi-norm p~(x')= supt(x, x')l, i t I.

xEAI

If B~ is a countable subset of B, dense in B for the semi-norm p~, then B ' = ~ B ~ is dense in B for the topology defined by {p~:iel} and the

cardinal of B' is not larger than ~, hence B is ~-separable for tr(A'), where A' = U At, and also for ~r(A"), where A" is the linear hull of A',

i e l

since tr(A)=tr(A"). B' is equicontinuous in E, hence in A"[dr]. The a(A")-dosure of B' is then equicontinuous in A" [6.1, so that B is equi- continuous in A" [dr]. Since A" is dense in A [o 'r] it results that B is equi- continuous in A[dr] and therefore dr and 6.(~) coincide in A. Q.E.D.

Corollary 1. (Grothendieck, [31). Let A be a separable set in a space E[oal . I f E[6rJ is a ~ space, then dr and 6*v, coincide on A.

Proof Since dr is coarser than 6*v, and drb* is coarser than 6.(,o) then, according to Theorem 1, 6* coincide with 6~b, on A. Q.E.D.

Corollary 2. Let ct be an infinite cardinal number. Given a subset A of a space E[o'rJ, let all be an uniformity on A x A compatible with the topology on A induced by dr. I f at/has a basis whose cardinal number is not larger than ~, then dr and drip) coincide on A.

Proof. Since dr~ is finer than 6* we must show that if P is an arbitrary closed set of A[dr(~)], then P is closed in A[6~]. Let x be a point of the closure of P in A[dr]. We can find a net {xd:deD} of points of P, which converges to x in A [$'1 so that the cardinal number of D is not larger than 0~. According to Theorem 1, 6* and 6*t~J coincide on the set {x} U {xa: d ~ D} and, therefore, the net {Xd:d e D} converges to x in A [#~)j, hence x ~ P. Q.E.D.

Corollary 3. (Grothendieck [31). Let A be a metrizable set in a space E[dr]. I f E[o'~J is a ~ space, then 8 and drb* coincide on A.

Corollary 4. Let ct be an infinite cardinal number. Let E be an ~t-barrelled, (~t-quasi-barreUed), space. I f in E there exists a family {A i : i e l } of precompact subsets, whose union is total in E, so that the

Locally Convex Spaces 147

cardinal number of I is not larger than ~, then E is barrelled, (quasi- barrelled).

Proof According to Theorem 1, the topology of E coincides with g('~. IfA is an absolutely convex bounded and closed, set of E'[a(E', E)], (E'[/~(E', E)]), then every subset of A, whose cardinal number is not larger than e, is equicontinuous in E, and, therefore, A is equicontinuous in E [gc,)] and, therefore, A is equicontinuous in E. Q.E.D.

The co-barrelled and co-quasi-barrelled spaces are introduced in [1] and [6]. In [6] it is given an example of a Mackey m-barrelled space, which is not barrelled. It is said that a space E is sequentially barrelled (quasi-barrelled), if given any bounded set A in E' [a(E' E),], (E'[fl(E', E)]), which is union of a countable family of equicontinuous sets, then A is equicontinuous, [4].

Theorem 2. Let ot and 7 be two infinite cardinal numbers, such that < 7. I f {Ei:i ~ 1} is a family of non-zero reflexive spaces, such that the

cardinal number of I is 7, then there is a dense subspace G in E = H {Ei: i e I} which satisfies the following conditions:

1. E'[#(E', G)] is an ~-barreIled space which is not 7-quasi-barrelled. 2. E'EI~(E', G)] is not sequentially quasi-barrelled space.

Proof. Let G be the subspace of E formed by all the points of E whose components are zero except a number ~ at most. If a bounded set M of G has a cardinal number which is not larger than ~, it is obvious that any point of the a(E, E')-closure of M has a number of non-zero components, which is not larger than ~, and, as E is re- flexive, E'[It(E', G)] is a-barrelled. Let x be a point of E, such that all their components {x~:ieI} are non-zero. Let P be the set of G, such that y ~ P if and only if xi = Yi for a finite number of components and y~ = 0 for the other those, being {y~: i e I} the components of y. The set P is bounded and its cardinal number is 7. The point x is in the a(E, E')- closure of P and x is not in G. Therefore P is not equicontinuous with respect to E'[#(E', G)]. Since G is barrelled, ([7], proof of Theorem 1), P is fl(G, E')-bounded and therefore, E'~(E ' , G)] is not y-quasi-barrelled. Finally it will be proved that E'[#(E', G)] is not sequentially quasi- barrelled. Let x be the same point just used. For every natural number n, let A, be the set of G, such that y ~ A, if and only if x~ = y~ for n components at most, and y~ = 0 for the other those. Obviously A, is

equicontinuous with respect to E'~(E ' , G)], n = 1, 2 . . . . . and P = ~) A,, n = l

hence E'[~(E', G)] is not sequentially quasi-barrelled. Q.E.D.

We are indebted to the referee by the proofs of Lemma and Theorem 1, respectively, more simple than the original ones.

148 M. Valdivia: Locally Convex Spaces

References

1. De Wilde, M., Houet, C.: On some permanence properties in topological vector spaces. (To be published.)

2. Garnir, H.G., De Wilde, M , Sehmets, J.: Analyse fonctionelle, I. Basel: Birkhliuser 1968.

3. Grothendieck, A.: Sur tes espaces (F) et (DF). Summa Brasil. Math. 3, 57--i23 (~954). 4. Husain, T.: Two new classes of locally convex spaces. Math. Ann. 166, 289--299 (1966). 5. Kothe, G.: Topological vector spaces I. Berlin-Heidelberg-New York: Springer 1969. 6. Levin, M., Saxon, S.: A note on lhe inheritance of properties of locally convex spaces

by subspaces of countable codimension. Proc. Amer. Math. Soc. 29, 97--102 (1971). 7. Valdivia, M.: On non bornological barrelled spaces. Ann. Inst. Fourier 22, 27--30 (1972).

Prof. Manuel Valdivia Facultad de Ciencias Paseo Valencia al Mar, 13 Valencia (Espana)

(Received December 20, 1971; in revised form February 16, I972)