# Krein-Smulian counterexample for the weak (not *) topology.

The Krein-Smulian Theorem states that for a convex set S, having weak*-closed intersections with closed balls implies being weak*-closed.

I would be really happy to have an example of a NOT weak-closed convex subset $S$ of a Banach space, such that the intersection with the closed balls are always weak-closed.

In this post, there are some counterexamples for the "convex" hypothesis. I would like to have counterexamples for the "weak*" hypothesis.

This is true for the weak topology too and it is easy to prove.

Proof. By the geometric form of the Hahn-Banach theorem, for a convex subset of $X$ to be closed is the same as to be weakly closed so it is enough to show that $S$ is closed.

Suppose that $S\subset X$ is convex and $S\cap B$ is (weakly) closed for every closed ball $B$ in $X$. Let $(x_n)_{n=1}^\infty$ be a sequence in $S$ which converges to some $x\in X$. Then $(x_n)_{n=1}^\infty$ is bounded, so it is contained in some ball $B$. However $S\cap B$ is closed, so $x\in S\cap B$. Consequently, $S$ is (weakly) closed. $\square$

• Thanks, Tomek! It seems Mazur's lemma is for sequences. But, is "sequentially weakly closed" the same as closed in all Banach spaces? Commented May 12, 2017 at 22:21
• So, my conclusion is that the difficulty in proving Krein-Smulian rests in the fact that when you have a strongly closed convex set $S \subset X^*$ and a point $\phi \in X^*$, it is not as easy to find $x \in X$ separating $S$ and $\phi$, as it is easy to apply Hahn-Banach and get an $F \in X^{**}$. Commented May 13, 2017 at 10:26
• This is so disappointing! :-P Commented May 13, 2017 at 10:30
• I'd like to take this opportunity and suggest to professors teaching the subject and to authors writing about it, that they introduce the theorem by first showing how easy it is to prove for the strong and weak topologies... and then, arguing that it is not so simple in the weak* case because Hahn-Banach cannot be directly applied. Thank you very much, Tomek! :-) Commented May 13, 2017 at 10:34
• I've never seen the Krein-Smulian Thm introduced with the easy case of the weak topology, but some texts give it as an exercise, e.g., Exc. 2.84 in R. E. Megginson: An introduction to Banach space theory, Springer 1998. doi.org/10.1007/978-1-4612-0603-3. This is a question about the bounded weak topology. For a review of such topologies, see Wrobel, A. J. (2020): Bounded topologies on Banach spaces and some of their uses in economic theory, arxiv.org/abs/2005.05202
– AJW
Commented Jul 29, 2020 at 18:42