Some aspects of the geometry of Lipschitz free spaces

Abstract : 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.
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Liste complète des métadonnées
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Submitted on : Friday, September 21, 2018 - 10:21:07 AM
Last modification on : Monday, September 24, 2018 - 5:14:06 PM
Document(s) archivé(s) le : Saturday, December 22, 2018 - 2:46:40 PM


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  • HAL Id : tel-01826406, version 2



Colin Petitjean. Some aspects of the geometry of Lipschitz free spaces. Functional Analysis [math.FA]. Université Bourgogne Franche-Comté, 2018. English. ⟨NNT : 2018UBFCD006⟩. ⟨tel-01826406v2⟩



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