A note on Banach spaces of Lipschitz functions

Abstract

This note is divided into two sections. The first establishes some properties of extreme Lipschitz functions that, it is hoped, will lead to satisfactory ways of characterizing them in general. The second section shows how ideas due to Lindenstrauss can be used to establish the existence of Lipschitz spaces that fail to be injective and fail the approximation property. Introduction. Our notation will follow essentially that of [6] and [11]. Given a metric space (S, rf), Li(S, d) denotes the Banach space of bounded real-valued functions on S with norm given by 11/11 = max(ll/im/ll,), where The closed subspace of functions / for which \f(s) -/()| = o(d(s, t)) is denoted by lip(S, d). If A CS, A denotes its complement in S, and if /: S->R is a function, M, denotes {s: \f(s)\ = ||/||oo}. In [11], Roy showed that a function / is an extreme point of the unit ball of Li(S, d), with 5 the unit interval and d the usual metric, if and only if I/ = 1 a.e. on M f and ||/|| = \\f%. = 1. (See [10]