A characterisation of L1-preduals in terms of extending Lipschitz maps

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

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On a global implicit function theorem for locally Lipschitz maps via non-smooth critical point theory

Quaestiones Mathematicae ◽

10.2989/16073606.2017.1391353 ◽

2017 ◽

Vol 41 (4) ◽

pp. 515-528 ◽

Cited By ~ 1

Author(s):

Marek Galewski ◽

Marius Rădulescu

Keyword(s):

Critical Point ◽

Implicit Function Theorem ◽

Critical Point Theory ◽

Point Theory ◽

Implicit Function ◽

Lipschitz Maps ◽

Locally Lipschitz

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Fuzzy Lipschitz maps and fixed point theorems in fuzzy metric spaces

Fuzzy Sets and Systems ◽

10.1016/j.fss.2009.05.009 ◽

2010 ◽

Vol 161 (8) ◽

pp. 1117-1130 ◽

Cited By ~ 3

Author(s):

Gabjin Yun ◽

Seungsu Hwang ◽

Jeongwook Chang

Keyword(s):

Fixed Point ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Fuzzy Metric Spaces ◽

Fuzzy Metric ◽

Lipschitz Maps

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Hermitian operators on Banach algebras of vector-valued Lipschitz maps

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2017.03.015 ◽

2017 ◽

Vol 452 (1) ◽

pp. 378-387 ◽

Cited By ~ 4

Author(s):

Osamu Hatori ◽

Shiho Oi

Keyword(s):

Banach Algebras ◽

Lipschitz Maps ◽

Vector Valued

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Fr ´Echet Differentiability Except For Γ‎-Null Sets

Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179) ◽

10.23943/princeton/9780691153551.003.0006 ◽

2012 ◽

Author(s):

Joram Lindenstrauss ◽

David Preiss ◽

Tiˇser Jaroslav

Keyword(s):

Banach Spaces ◽

Common Point ◽

Lipschitz Functions ◽

Infinite Dimensional ◽

Lipschitz Maps ◽

Almost Everywhere ◽

Fréchet Differentiability ◽

Gateaux Derivatives ◽

Frechet Differentiability ◽

Porous Sets

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

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