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Nonlinear Geometry of Banach Spaces

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Topics in Banach Space Theory

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 233 ))

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Abstract

A Banach space is, by its nature, also a metric space. When we identify a Banach space with its underlying metric space, we choose to forget its linear structure. The fundamental question of nonlinear geometry is to determine to what extent the metric structure of a Banach space already determines its linear structure. What can be said of two Banach spaces that are Lipschitz-isomorphic? uniformly homeomorphic? or coarsely Lipschitz-isomorphic in the spirit of M. Gromov’s geometric theory of groups? These questions are still partly open, and investigating them requires some quite technical tools. In this chapter, we will consider only basic techniques whose purpose is to produce specific linear maps from nonlinear ones (typically, Lipschitz maps).

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  1. Departamento de Matemáticas, Universidad Pública de Navarra, Pamplona, Spain

    Fernando Albiac & Nigel J. Kalton

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  1. Fernando Albiac
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  2. Nigel J. Kalton
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Albiac, F., Kalton, N.J. (2016). Nonlinear Geometry of Banach Spaces. In: Topics in Banach Space Theory. Graduate Texts in Mathematics, vol 233 . Springer, Cham. https://doi.org/10.1007/978-3-319-31557-7_14

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