Etude des Espaces Lipschitz-libres

Abstract : Godefroy and Ozawa have proved that there exists a compact space with a free space failing the approximation property. Then it is natural to ask what are the metric spaces whose freespace has the bounded approximation property. Grothendieck has proved that a separable Banach space with the approximation property has the metric approximation property. This result justifies why it is interesting to know whether a free space is a dual space. The first chapter is dedicated to duality. First we introduce a result to prove that a Banach space is a dual space, under some conditions. Then we explain how to use it in the context offree spaces and finally we apply it to countable or ultrametric proper metric spaces.In the second chapter, we study the metric approximation property of free spaces overcountable proper metric spaces.In the third chapter, ultrametric spaces are investigated. We prove first that the free spaceover a proper ultrametric space has the metric approximation property, is isomorphic to l1 andadmits a predual isomorphic to c0. Finally, in collaboration with P. Kaufmann et A. Proch`azka,we prove that the free space over a ultrametric space is never isometric to l1 and we generalizethis result to some subsets of separable R-trees.
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Aude Dalet. Etude des Espaces Lipschitz-libres. Analyse fonctionnelle [math.FA]. Université de Franche-Comté, 2015. Français. ⟨NNT : 2015BESA2050⟩. ⟨tel-01680253⟩

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