Lipschitz-free Banach spaces

Volume 159 / 2003

G. Godefroy, N. J. Kalton Studia Mathematica 159 (2003), 121-141 MSC: Primary 46B20; Secondary 46B26, 46B28. DOI: 10.4064/sm159-1-6


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.


  • G. GodefroyÉquipe d'Analyse
    Université Paris VI
    Boîte 186
    4, Place Jussieu
    75252 Paris Cedex 05, France
  • N. J. KaltonDepartment of Mathematics
    University of Missouri-Columbia
    Columbia, MO 65211, U.S.A.

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