PreprintarXiv licencearXiv
On complementation and L-embedding of Lipschitz-free spaces in their biduals
Abstract
A closed subspace E of a Banach space F is \emph{L-embedded in F} if there is a projection P: F→ E such that |x|=|Px|+|x−Px| for all x∈ F. We show that if X is a proper, geodesic metric space with locally finite H1-measure, then its Lipschitz-free space, denoted by F(X), is L-embedded in its bidual. We also extend an argument by Cúth, Kalenda, and Kaplický, to show that if E is a finite dimensional normed space and C≥ 1 is such that every subspace of E is C-complemented in E, then cal F(E) is C-complemented in its bidual. In particular, for every two-dimensional normed space E, cal F(E) is $1$-complemented in its bidual.
Related
- On the L-embeddability of Lipschitz-free spaces in their biduals
- Isometric copies of ℓ∞n in Lipschitz-free spaces over finite metric spaces
- Lipschitz-free spaces over separable Banach spaces and their complementability in the bidual
- The Ball-Covering Property in Lipschitz-Free Spaces
- The Dunford-Pettis property for Lipschitz-free spaces over ℝn
BibTeX
@misc{braga2026complementation,
title = {{On complementation and $L$-embedding of Lipschitz-free spaces in their biduals}},
author = {Bruno M. Braga and Chris Gartland and Gilles Lancien and Eva Pernecká and Pavlos Motakis and Thomas Schlumprecht},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18608},
eprint = {2610.18608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18608},
}
RIS
TY - UNPB
TI - On complementation and L-embedding of Lipschitz-free spaces in their biduals
AU - Bruno M. Braga
AU - Chris Gartland
AU - Gilles Lancien
AU - Eva Pernecká
AU - Pavlos Motakis
AU - Thomas Schlumprecht
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18608
UR - https://arxiv.org/abs/2610.18608
AB - A closed subspace E of a Banach space F is \emph{L-embedded in F} if there is a projection P: F→ E such that |x|=|Px|+|x−Px| for all x∈ F. We show that if X is a proper, geodesic metric space with locally finite H^1-measure, then its Lipschitz-free space, denoted by F(X), is L-embedded in its bidual. We also extend an argument by Cúth, Kalenda, and Kaplický, to show that if E is a finite dimensional normed space and C≥ 1 is such that every subspace of E is C-complemented in E, then cal F(E) is C-complemented in its bidual. In particular, for every two-dimensional normed space E, cal F(E) is $1$-complemented in its bidual.
ER -