Some aspects of the geometry of Lipschitz free spaces.First and foremost, we give the fundamental properties of Lipschitz free spaces. Then, we prove that the canonical image of a metric space M is weakly closed in the associated free space F(M). We prove a similar result for the set of molecules.In the second chapter, we study the circumstances in which F(M) is isometric to a dual space. In particular, we generalize a result due to Kalton on this topic. Subsequently, we focus on uniformly discrete metric spaces and on metric spaces originating from p-Banach spaces.In the next chapter, we focus on l1-like properties. Among other things, we prove that F(M) has the Schur property provided the space of little Lipschitz functions is 1-norming for F(M). Under additional assumptions, we manage to embed F(M) into an l1-sum of finite dimensional spaces.In the fourth chapter, we study the extremal structure of F(M). In particular, we show that any preserved extreme point in the unit ball of a free space is a denting point. Moreover, if F(M) admits a predual, we obtain a precise description of its extremal structure.The fifth chapter deals with vector-valued Lipschitz functions.We generalize some results obtained in the first three chapters.We finish with some considerations of norm attainment. For instance, we obtain a density result for vector-valued Lipschitz maps which attain their norm.
Author(s): Petitjean, C.
Publisher: Université de Bourgogne
Year: 2018
Language: English
Pages: 124
City: Dijon
1. Résumé de la thèse......Page 5
2. Summary of the thesis......Page 13
3. Notation......Page 20
Definition and first properties......Page 23
Weak closure of delta(M)......Page 26
Weak closure of the set of molecules......Page 27
Perspectives......Page 29
The spaces of little Lipschitz functions......Page 31
Natural preduals......Page 34
The uniformly discrete case......Page 37
Metric spaces originating from p-Banach spaces......Page 42
Perspectives......Page 45
The Schur property......Page 47
Quantitative versions of the Schur property......Page 49
Embeddings into l1-sums......Page 51
Metric spaces originating from p-Banach spaces......Page 53
Perspectives......Page 54
Extremal structure of Lipschitz free spaces......Page 57
General results......Page 60
Extremal structure for spaces with natural preduals......Page 65
The uniformly discrete case......Page 67
Proper metric spaces......Page 69
Perspectives......Page 71
Tensor products......Page 73
Vector-valued Lipschitz free spaces......Page 75
Duality results......Page 78
Schur properties in the vector-valued case......Page 83
Norm attainment......Page 88
Perspectives......Page 91
The conjecture......Page 93
Origin of the conjecture......Page 96
Preliminary results......Page 97
Proof of the main result......Page 101
Characterization of polytopes in R1.......Page 102
Characterization of polytopes in R2.......Page 104
Construction of R3 and conclusion.......Page 106
Weakening assumption (H2)......Page 107
Introduction......Page 109
Property Q and Kalton's graphs......Page 110
The James space......Page 111
Kalton's graphs do not embed into the James space......Page 112
