JournalsrmiVol. 22, No. 1pp. 93–110

An extension of the Krein-Šmulian Theorem

  • Antonio S. Granero

    Universidad Complutense de Madrid, Spain
An extension of the Krein-Šmulian Theorem cover
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Let XX be a Banach space, uXu\in X^{**} and K,ZK, Z two subsets of XX^{**}. Denote by d(u,Z)d(u,Z) and d(K,Z)d(K,Z) the distances to ZZ from the point uu and from the subset KK 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 KXK\subset X^{**} such that d(K,X)=0d(K,X)=0 satisfies d(cow(K),X)=0d(\overline{\text{co}}^{w^*}(K),X)=0. We extend this result in the following way: if ZXZ\subset X is a closed subspace of XX and KXK\subset X^{**} is a w^*-compact subset of XX^{**}, then

d(cow(K),Z)5d(K,Z).d(\overline{\text{co}}^{w^*}(K),Z)\leq 5 d(K,Z).

Moreover, if ZKZ\cap K is w^*-dense in KK, then d(cow(K),Z)2d(K,Z)d(\overline{\text{co}}^{w^*}(K),Z)\leq 2 d(K,Z). However, the equality d(K,X)=d(cow(K),X)d(K,X)=d(\overline{\text{co}}^{w^*}(K),X) holds in many cases, for instance, if 1⊈X\ell_1\not\subseteq X^*, if XX has w^*-angelic dual unit ball (for example, if XX is WCG or WLD), if X=1(I)X=\ell_1(I), if KK is fragmented by the norm of XX^{**}, etc. We also construct under CHCH a w^*-compact subset KB(X)K\subset B(X^{**}) such that KXK\cap X is w^*-dense in KK, d(K,X)=12d(K,X)=\frac 12 and d(cow(K),X)=1d(\overline{\text{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