A Note on Differentiability of Lipschitz Maps

Rafał Górak

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We show that there is a universal control on the Szlenk index of a Lipschitz-quotient of a Banach space with countable Szlenk index. It is in particular the case when two Banach spaces are Lipschitz-homeomorphic. This provides information on the Cantor index of scattered compact sets $K$ and $L$ such that $C (L)$ is a Lipschitz-quotient of $C (K)$ (that is the case in particular when these two spaces are Lipschitz-homeomorphic). The proof requires tools of descriptive set theory.