The extension of the Krein-Šmulian theorem for order-continuous Banach lattices
Volume 79 / 2008
Banach Center Publications 79 (2008), 79-93
MSC: 46B20, 46B26.
DOI: 10.4064/bc79-0-6
Abstract
If 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).
