An extension of the Krein-Šmulian Theorem

Abstract

Let $X$ be a Banach space, $u\in X^{\ast \ast }$ and $K,Z$ two subsets of $X^{\ast \ast }$. 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${}^{\ast }$-compact subset $K\subset X^{\ast \ast }$ such that $d(K,X)=0$ satisfies $d({\overline{\text{co}}}^{w^{\ast }}(K),X)=0$. We extend this result in the following way: if $Z\subset X$ is a closed subspace of $X$ and $K\subset X^{\ast \ast }$ is a w${}^{\ast }-$compact subset of $X^{\ast \ast }$, then

Moreover, if $Z\cap K$ is w${}^{\ast }$-dense in $K$, then $d({\overline{\text{co}}}^{w^{\ast }}(K),Z)\le 2d(K,Z)$. However, the equality $d(K,X)=d({\overline{\text{co}}}^{w^{\ast }}(K),X)$ holds in many cases, for instance, if ${\ell }_1⊈X^{\ast }$, if $X$ has w${}^{\ast }$-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 $X^{\ast \ast }$, etc. We also construct under $CH$ a w${}^{\ast }$-compact subset $K\subset B(X^{\ast \ast })$ such that $K\cap X$ is w${}^{\ast }$-dense in $K$, $d(K,X)=\frac{1}{2}$ and $d({\overline{\text{co}}}^{w^{\ast }}(K),X)=1$.