# Closed convex bounded sets are weakly compact for which spaces?

It is known that for all reflexive Banach spaces, closed convex bounded sets are weakly compact (compact for the weak topology).

What is the general class of topological vector spaces for which this is true ? For example, is it true that for all reflexive complete locally convex topological vector spaces, closed convex bounded sets are weakly compact ?

• Hence my question – Jon-S Oct 10 '16 at 2:08
• There is a beaiful theorem by Davis-Johnson-Pelczynski in" Factoring weakly compact operators", which is available freely online in sciencedirect.com/science/article/pii/0022123674900445 which states that every weakly compact operator between Banach spaces factors through a refelxive space. – Uri Bader Oct 10 '16 at 6:31
• You should read remark 2 in this paper, and the following discussion. It provides a positive answer for Frechet spaces, I believe. – Uri Bader Oct 10 '16 at 6:32

It is well-known that a Hausdorff locally convex space is semi-reflexive (i.e., the canonical map into its bidual is surjective) if and only if every weakly closed bounded set is weakly compact. This is proposition 23.18 in Introduction to Functional Analysis of Meise and Vogt.

(It is mainly a consequence of Alaoglu's theorem.)

• I was all into writing my answer and didn't realize you posted this already. In any case my effort wasn't for nothing: I learned something new. – Uri Bader Oct 10 '16 at 8:01

The following is from Shaefer's "Topological Vector Spaces", sections 5.5 and 5.6.

For a locally convex (Hausdorff) $E$, the injection into its bidual is surjective iff every bounded set in $E$ is weakly-relatively-compact. In this case $E$ is said to be semi-refelexive. The map into the bidual need not be continuous. This bijection is continuous iff $E$ is also barreled. In this case this bijection is in fact an isomorphism of topological vector spaces and $E$ is said to be reflexive.

A general topological vector space which injects into its bidual, when taken with the weak topology, is clearly locally convex Hausdorff, so we can safely restrict ourselves to this class.

So, the answer to the first question is "the general class of locally convex Hausdorff topological vector spaces for which this is true is called semi-reflexive spaces" and the answer to the second is "yes, in particular it is true for reflexive spaces".

• No, bounded sets in locally convex spaces are always weakly precompact. You have to replace precompactness by relative compactness to obtain a characterization of reflexivity. – Jochen Wengenroth Oct 10 '16 at 8:28
• @JochenWengenroth, Thanks for the correction. I fixed that typo. – Uri Bader Oct 10 '16 at 12:21
• @JochenWengenroth: What is the difference between "precompact" and "relatively compact"? I always thought they were synonyms, both meaning "the closure is compact", and Wikipedia thinks so too. – Nate Eldredge Oct 10 '16 at 13:07
• @NateEldredge Usually I use these as synonym as well and nothing bad happens, but they have different meanings for peopel in TVS. "Relatively compact" is a property of a subset in a topological space saying that its closure is compact. "Precompact" is a property of a uniform space saying that its completion is compact. For a subset of a complete uniform space (eg a TVS) they coinside. – Uri Bader Oct 10 '16 at 14:24
• Thanks @JochenWengenroth. So what I wrote two steps above, the sentence ending by (I think) and the sentence after, should have regarded the unit ball in a Banach space, not the whole space, right? – Uri Bader Oct 10 '16 at 14:40