Porosity in the space of H{ö}lder-functions

Abstract

Let (X, d) be a bounded metric space with a base point 0 X , (Y, $\bullet$) be a Banach space and Lip $\alpha$ 0 (X, Y) be the space of all $\alpha$-H{ö}lderfunctions that vanish at 0 X , equipped with its natural norm (0 < $\alpha$ $\le$ 1). Let 0 < $\alpha$ < $\beta$ $\le$ 1. We prove that Lip $\beta$ 0 (X, Y) is a $\sigma$-porous subset of Lip $\alpha$ 0 (X, Y), if (and only if) inf{d(x, x ') : x, x ' $\in$ X; x = x ' } = 0 (i.e. d is non-uniformly discrete). A more general result will be given.