# An extension of the Krein-Šmulian Theorem

### Antonio S. Granero

Universidad Complutense de Madrid, Spain

## Abstract

Let $X$ be a Banach space, $u∈X_{∗∗}$ 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 $K⊂X_{∗∗}$ such that $d(K,X)=0$ satisfies $d(co_{w_{∗}}(K),X)=0$. We extend this result in the following way: if $Z⊂X$ is a closed subspace of $X$ and $K⊂X_{∗∗}$ is a w$_{∗}−$compact subset of $X_{∗∗}$, then

Moreover, if $Z∩K$ is w$_{∗}$-dense in $K$, then $d(co_{w_{∗}}(K),Z)≤2d(K,Z)$. However, the equality $d(K,X)=d(co_{w_{∗}}(K),X)$ holds in many cases, for instance, if $ℓ_{1}⊆X_{∗}$, if $X$ has w$_{∗}$-angelic dual unit ball (for example, if $X$ is WCG or WLD), if $X=ℓ_{1}(I)$, if $K$ is fragmented by the norm of $X_{∗∗}$, etc. We also construct under $CH$ a w$_{∗}$-compact subset $K⊂B(X_{∗∗})$ such that $K∩X$ is w$_{∗}$-dense in $K$, $d(K,X)=21 $ and $d(co_{w_{∗}}(K),X)=1$.

## Cite this article

Antonio S. Granero, An extension of the Krein-Šmulian Theorem. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 93–110

DOI 10.4171/RMI/450