The extension of the Krein-Šmulian theorem for order-continuous Banach lattices
Volume 79 / 2008
Abstract
If $X$ is a Banach space and $C\subset X$ a convex subset, for $x^{**}\in X^{**}$ and $A\subset X^{**}$ let $d(x^{**},C)=\inf \{\|x^{**}-x\| : x\in C\}$ be the distance from $x^{**}$ to $C$ and $\hat d(A,C)=\sup \{d(a,C):a\in A\}$. Among other things, we prove that if $X$ is an order-continuous Banach lattice and $K$ is a w$^*$-compact subset of $X^{**}$ we have: (i) $\hat d(\overline {{\rm co}} ^{w^*}(K),X)\leq 2\hat d(K,X)$ and, if $K\cap X$ is w$^*$-dense in $K$, then $\hat d(\overline {{\rm co}} ^{w^*}(K),X) =\hat d(K,X)$; (ii) if $X$ fails to have a copy of $\ell _1(\aleph _1)$, then $\hat d(\overline {{\rm co}} ^{w^*}(K),X) =\hat d(K,X)$; (iii) if $X$ has a 1-symmetric basis, then $\hat d(\overline {{\rm co}} ^{w^*}(K),X) =\hat d(K,X)$.
