Finitely additive measures and complementability of Lipschitz-free spaces

Abstract

We prove in particular that the Lipschitz-free space over a finitely-dimensional normed space is complemented in its bidual. For Euclidean spaces the norm of the respective projection is $1$. As a tool to obtain the main result we establish several facts on the structure of finitely additive measures on finitely-dimensional spaces.

arXiv comment24 pages; we corrected some misprints, reorganized a bit the introduction and added few comments (in particular one on the Leray projection)

References in corpus (5)

• On the structure of Lipschitz-free spaces
• Lipschitz-free spaces over ultrametric spaces
• The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of ${\mathbb{R}}^N$
• Isometric representation of Lipschitz-free spaces over convex domains in finite-dimensional spaces
• Some natural subspaces and quotient spaces of $L^1$

Cited by in corpus (4)

• Integral representation and supports of functionals on Lipschitz spaces
• De Leeuw representations of functionals on Lipschitz spaces
• Normal functionals on Lipschitz spaces are weak${}^{\ast }$ continuous
• Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces