The class of universally Krein-Šmulian Banach spaces

Abstract

If A and B are subsets of a Banach space X, let d(A, B) = sup{inf{‖a − b‖ : b ∈ B} : a ∈ A} denote the distance from A to B. In this paper we study when the distances d(cow ∗ (K), Y ) are M controlled by the distances d(K, Y ) (that is, d(cow ∗ (K), Y ) ≤ Md(K, Y ) for some 1 ≤ M < ∞), Y being a subspace of a dual Banach space X∗ and K a weak∗-compact subset of X∗. Among other things, we show that for a Banach space Y the following are equivalent: (i) Y fails to have a copy of `1(c); (ii) Y is universally Krein-Smulian (that is, Y satisfies the Krein-Smulian Theorem with respect to the topology σ(Y, Z), for every norming subspace Z of Y ∗);(iii) Y has control (in fact 3-control) inside every dual Banach space X∗ that contains Y as a subspace. Moreover, Y contains a copy of `1(c) if and only if there exists a dual Banach space X∗ such that for every constant 1 ≤ M < ∞ there exists a subspace ZM of X∗ isomorphic to Y such that (ZM , w∗) satisfies the Krein-Smulian Theorem but ZM fails to have M -control inside X ∗.