Porosity in the space of Hölder-functions

2021/05/11 by Bachir, Mohammed
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2105.04849

Abstract

Let (X, d) be a bounded metric space with a base point 0 X , (Y, \bullet) 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) infd(x, x ') : x, x ' ∈ X; x = x ' = 0 (i.e. d is non-uniformly discrete). A more general result will be given.

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