$\overline{\text{conv}}\{x_n: n\in \mathbb{N}\}$ has empty interior for a w-conv seqeunce $(x_n)_n$.

Question

Given $x_n \to x$ weakly in a Banach space X, $\dim X = \infty$.

Show that $\text{int}\left\{ \overline{\text{conv}}\{x_n:n\in\mathbb N \}\right\}$ is empty.

Can anyone help me with this problem? Any hints? Prob. Krein-Millmann is needed but I don't see how to use it.

Answer 1

Without loss of generality we may assume that the set contains $\overline{\text{conv}}\{x_n:n\in\mathbb N \}$ a closed unit ball $B.$ Since the set $\overline{\text{conv}}\{x_n:n\in\mathbb N \}$ is weakly compact thus the unit ball $B$ is weakly compact as weakly closed subset of weakly compact set. Thus $X$ is reflexive and every weakly compact subset of $X$ is cotained in closed convex hull of weakly null sequence. But the last statement contradicts to this result: http://math.slu.edu/~freeman/wGschurJFA.pdf .