An extension of the Krein-Šmulian Theorem

Abstract

Let $X$ be a Banach space, $u\in X^{**}$ and $K, Z$ two subsets of $X^{**}$. Denote by $d(u,Z)$ and $d(K,Z)$ the distances to $Z$ from the point $u$ and from the subset $K$ respectively. The Krein-Šmulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w$^*$-compact subset $K\subset X^{**}$ such that $d(K,X)=0$ satisfies $d(\overline{\text{co}}^{w^*}(K),X)=0$. We extend this result in the following way: if $Z\subset X$ is a closed subspace of $X$ and $K\subset X^{**}$ is a w$^*-$compact subset of $X^{**}$, then $$ d(\overline{\text{co}}^{w^*}(K),Z)\leq 5 d(K,Z). $$ Moreover, if $Z\cap K$ is w$^*$-dense in $K$, then $d(\overline{\text{co}}^{w^*}(K),Z)\leq 2 d(K,Z)$. However, the equality $d(K,X)=d(\overline{\text{co}}^{w^*}(K),X)$ holds in many cases, for instance, if $\ell_1\not\subseteq X^*$, if $X$ has w$^*$-angelic dual unit ball (for example, if $X$ is WCG or WLD), if $X=\ell_1(I)$, if $K$ is fragmented by the norm of $X^{**}$, etc. We also construct under $CH$ a w$^*$-compact subset $K\subset B(X^{**})$ such that $K\cap X$ is w$^*$-dense in $K$, $d(K,X)=\frac 12$ and $d(\overline{\text{co}}^{w^*}(K),X)=1$.