Compactness and Weak Sequential Compactness: The Eberlein-\v{S}mulian Theorem

In a general topological space that is not metrizable a sequence may converge to more than one point. Show that this cannot occur for the $W$-weak topology on a normed linear space $X$, where $W$ is a subspace of $X^{*}$ that separates points in $X$.

Show that there is a bounded sequence in $L^{\infty }[0,1]$ that fails to have a weakly convergent subsequence. Show that the closed unit ball of $C[a, b]$ is not weakly compact.

Let $K$ be a compact metric space with infinitely many points. Show that there is a bounded sequence in $C(K)$ that fails to have a weakly convergent subsequence (see Problem 11), but every bounded sequence of continuous linear functionals on $C(K)$ has a subsequence that converges pointwise to a continuous linear functional on $C(K)$.