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ISOMETRIC REPRESENTATION OF LIPSCHITZ-FREE SPACES OVER CONVEX DOMAINS IN FINITE-DIMENSIONAL SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices Real functions

Published online by Cambridge University Press:  04 April 2017

Marek Cúth ,
Ondřej F. K. Kalenda  and
Petr Kaplický
Marek Cúth
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic email [email protected]
Ondřej F. K. Kalenda
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic email [email protected]
Petr Kaplický
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic email [email protected]
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Abstract

Let $E$ be a finite-dimensional normed space and $\unicode[STIX]{x1D6FA}$ a non-empty convex open set in $E$. We show that the Lipschitz-free space of $\unicode[STIX]{x1D6FA}$ is canonically isometric to the quotient of $L^{1}(\unicode[STIX]{x1D6FA},E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$.

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 26A16: Lipschitz (Hölder) classes
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 538 - 552
Copyright
Copyright © University College London 2017 

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