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Question

Are bounded products of barrelled normed spaces barrelled? Oren Kolman

Question (Lurje [1]; Drewnowski, Florencio, and Paul [2]):

Suppose that S is a set and E is a barrelled normed vector space. Is l_\infty(S, E) barrelled?

Drewnowski, Florencio, and Paul [2] study this problem and prove that the answer is positive, provided that at least one of the cardinals |S| and |E| is not an Ulam-measurable cardinal. This note summarises their results and provides the definitions necessary to understand the question.

Let S be a set and E be a locally convex vector space. The space l_\infty(S, E) is the locally convex space of all bounded functions from S to E equipped with the topology of uniform convergence on S. For present purposes, a space is a locally convex Hausdorff topological vector space.

Every locally convex Hausdorff topological vector space has a base of neighbourhoods of the origin formed by barrels. A space E is barrelled if every barrel contains a neighbourhood of 0. For example, every Banach space is barrelled.

A barrel B 'subset' E is a closed, convex, balanced, absorbing set.

A set C 'subset' E is convex if ( 'for all'x, y \in C)( 'for all't \in [0, 1])( tx + (1 - t)y \in C).

A set D 'subset' E is balanced if aD 'subset' D whenever |a| <= 1.

A convex set A 'subset' E is absorbing if ( 'for all'x \in E)( 'exists't = t(x) > 0)( x \in tA).

There are several equivalent characterisations of barrelled spaces (see e.g. Ferrando, López Pellicer, Sánchez Ruiz [3] for definitions and proofs):

Theorem 1. Suppose E is a locally convex space. The following conditions are equivalent:
(1.1) E is barrelled;
(1.2) for every space F, each pointwise bounded subset H of L(E, F) is equicontinuous;
(1.3) every linear mapping T from E into any space F is nearly continuous;
(1.4) given any Frechet space F, if T: E --> F is a linear mapping with a closed graph in E ×F then T is continuous;
(1.5) given any Banach space F, T: E --> F is a linear mapping with a closed graph in E×F; then T is continuous;
(1.6) E is Banach-Mackey and quasi-barrelled.

The main results bearing on the question above are the following:

Theorem 2 (Lurje [1]). If E is a barrelled normed space, then l_\infty(\omega, E ) is barrelled.

Theorem 3 (Drewnowski, Florencio, Paul [2]). If |S| or |E| is not Ulam-measurable, and E is a barrelled normed space, then l_\infty(S, E ) is barrelled.

Definition. A cardinal \kappa is Ulam-measurable if there is a non-principal \sigma-complete ultrafilter on \kappa.

Ulam-measurable cardinals are sometimes called \omega-measurable; the least \omega-measurable cardinal \kappa is measurable: there is a \kappa-complete non-principal ultrafilter on \kappa. The axiom of constructibility V = L implies that there are no measurable cardinals. Hence ordinary set theory (ZFC), if consistent, will never produce a non-barrelled product l_\infty(S, E ).

Corollary. V = L implies that l_\infty(S, E )is barrelled for every set S and every barrelled normed space E.

So it is consistent relative to ZFC that every bounded product of barrelled normed spaces is barrelled. The question is thus whether it is a theorem of ZFC that bounded products of barrelled normed spaces preserve barrelledness, or whether a large cardinal axiom will enable one to find a counterexample.

There is also a characterisation of barrelled products in terms of ultrapowers.

Theorem 4 (Drewnowski, Florencio, Paul [2]). Let S be a set and E be a barrelled normed space. The following are equivalent:
(4.1) l_\infty(S, E );
(4.2) for every non-principal ultrafilter U over S, the ultrapower (E)_U is barrelled.

References

[1] P. Lurje, Tonnelierheit in lokalkonvexen Vektorgruppen, Manuscripts Math. 14 (1974), 107-121.

[2] L. Drewnowski, M. Florencio, P.J. Pa\'ul, On the barrelledness of spaces of bounded vector functions, Arch. Math. 63 (1994), 449-458.

[3] J.C. Ferrando, M. López Pellicer, L.M. Sánchez Ruiz, Metrizable barrelled spaces, Pitman Reseach Notes in Mathematics Series, volume 332, Longman/John Wiley & Sons, Harlow and New York, 1995.