Lipschitz-free Banach spaces

G. Godefroy ¹   ; N. J. Kalton ²

¹ Équipe d'Analyse Université Paris VI Boîte 186 4, Place Jussieu 75252 Paris Cedex 05, France
² Department of Mathematics University of Missouri-Columbia Columbia, MO 65211, U.S.A.

Studia Mathematica, Tome 159 (2003) no. 1, pp. 121-141

Résumé

We show that when a linear quotient map to a separable Banach space $X$ has a Lipschitz right inverse, then it has a linear right inverse. If a separable space $X$ embeds isometrically into a Banach space $Y$, then $Y$ contains an isometric linear copy of $X$. This is false for every nonseparable weakly compactly generated Banach space $X$. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space $X$ has the bounded approximation property and $Y$ is Lipschitz isomorphic to $X$, then $Y$ has the bounded approximation property.

Détail

DOI : 10.4064/sm159-1-6

Mots-clés : linear quotient map separable banach space has lipschitz right inverse has linear right inverse separable space embeds isometrically banach space contains isometric linear copy false every nonseparable weakly compactly generated banach space canonical examples nonseparable banach spaces which lipschitz isomorphic linearly isomorphic constructed banach space has bounded approximation property lipschitz isomorphic has bounded approximation property