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 .