POROSITY IN THE SPACE OF HÖLDER-FUNCTIONS

Let (X, d) be a bounded metric space with a base point 0 X , (Y, •) be a Banach space and Lip α 0 (X, Y) be the space of all α-Hölderfunctions that vanish at 0 X , equipped with its natural norm (0 < α ≤ 1). Let 0 < α < β ≤ 1. We prove that Lip β 0 (X, Y) is a σ-porous subset of Lip α 0 (X, Y), if (and only if) inf{d(x, x ′) : x, x ′ ∈ X; x = x ′ } = 0 (i.e. d is non-uniformly discrete). A more general result will be given