Extensions of smooth mappings into biduals and weak continuity

Abstract

Our work is based on a number of tools that are of independent interest. We prove, for every pair of Banach spaces X, Y, that any continuous mapping T : B-X -> Y, which is uniformly differentiable of order up to k in the interior of B-X, can be extended, preserving its best smoothness, into a bidual mapping (T) over tilde : B-X** -> Y**. Another main tool is a result of Zippin's type. We show that weakly Cauchy sequences in X = C(K) can be uniformly well approximated by weakly Cauchy sequences from a certain C[0, alpha], alpha is a countable ordinal, subspace of X**. (C) 2012 Elsevier Inc. All rights reserved.