2

Let $E$ be an infinite-dimensional Banach space and $E^*$ its dual. If $A$ is a subset of $E$ and $x$ is a weak limit point of $A$, i.e. every weakly open deleted neighborhood of $x$ intersects $A$, can we always find a sequence $x_n\in A$ that converges weakly to $x$?

This property is enjoyed by any metric spaces, as we can determine how large a neighborhood is using the metric. But the weak topology is known to be not metrizable, and the answer to this question becomes less evident.

Is there a proof or a counterexample?

trisct
  • 5,373
  • 1
    Here is a counterexample. – Daniel Fischer Dec 01 '19 at 14:16
  • If we add some settings of compactness then things will work. There is an exercise on Rudin's Functional Analysis (chap 3) that I'll adapt here. Let $A$ be compact and suppose $x \in A$ is a weak limit point of some countable set $E \subset A$. Then there is a sequence ${x_n}$ in $E$ that converges weakly to $x$. – Degenerate D May 09 '25 at 18:09

0 Answers0