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arXiv · 2105.04849

Porosity in the space of H{\"o}lder-functions

Mohammed Bachir
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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{\"o}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.

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Mohammed Bachir. 2021-05-11. Porosity in the space of H{\"o}lder-functions. https://arxiv.org/abs/2105.04849

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