The space of Lipschitz maps from a compactum to a locally convex set

Abstract

Let X be a non-discrete metric compactum and Y a separable locally compact, locally convex set without isolated points in a normed linear space. The spaces of continuous maps and Lipschitz maps from X to Y are denoted by C( X, Y) and LIP( X, Y), respectively. For each k>0, k-LIP( X, Y) and LIP k ( X, Y) denote the subspaces of all f; ϵ LIP( X, Y) with the Lipschitz constant lip f; ⩽ k and lip f< k, respectively. Here is proved that ( C( X, Y), LIP( X, Y)) is an ( s, Σ)-manifold pair and each LIP k ( X, Y) is a Σ-manifold, where s = (−1, 1) ω is the pseudo-interior of the Hilbert cube Q = [−1, 1] ω and Σ = {( x i ) ϵ Q| sup| x i |<1}. In general, k-LIP( X, Y) need not be a Q-manifold. However, in case Y is open in its convex hull, it is proved that ( k-LIP( X, Y), LIP k ( X, Y)) is a ( Q, Σ)-manifold pair for each k>0.