The non-linear geometry of Banach spaces after Nigel Kalton
type: Item
instance of: scholarly article
title: The non-linear geometry of Banach spaces after Nigel Kalton
zbMATH Open document ID: 1317.46016
DOI: 10.1216/RMJ-2014-44-5-1529
author: Gilles Godefroy
author: G. Lancien
author: V. Zizler
published in: Rocky Mountain Journal of Mathematics
publication date: 2015-01-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.2958
full work available at URL: https://projecteuclid.org/euclid.rmjm/1420071553
review text: Roughly speaking, ``nonlinear geometry of Banach spaces'' stands for a theory that aims at studying the linear structure of Banach spaces by means of nonlinear maps, e.g., Lipschitz or uniformly continuous maps. The survey paper under review presents a number of the deep results in this area obtained by the late Nigel Kalton, some of which were published in his posthumous papers, e.g., [\textit{N.J.~Kalton}, Fundam.\ Math.\ 212, No. 1, 53--69 (2011; Zbl 1220.46014), Math.\ Ann.\ 354, No. 4, 1247--1288 (2012; Zbl 1268.46018), Trans.\ Am.\ Math.\ Soc.\ 365, No. 2, 1051--1079 (2013; Zbl 1275.46011), Isr.\ J. Math.\ 194, Part A, 151--182 (2013; Zbl 1275.46012)]. Several theorems are explained with complete proofs, while other proofs are just sketched, but in a highly illuminating way. The paper starts with some tables indicating between which classical Banach spaces nonlinear embeddings of a special type are possible or not possible. The third section studies asymptotic uniform smoothness and convexity in connection with nonlinear mappings. The subject matter of Section~4 is the characterisation of Banach space properties by means of the embeddability of certain metric spaces (especially graphs), i.e., to describe the linear structure of a Banach space through its structure as a metric space, which has become known as the Ribe programme after the seminal contributions by M.~Ribe in the 1970s. In Section~5 the authors take a look at nonseparable spaces, while they give an introduction to the Novikov conjecture and its relation to coarse embeddings in Section~6. This section is especially designed for the lay reader in these matters and will be much appreciated by such readers, like myself. Finally, in Section~7 the Lipschitz free space, also known as the Arens-Eells space, of a metric or Banach space is presented; this is the canonical predual of the space of Lipschitz functions on the metric space and can be understood as its linearisation. Throughout the text, several …
reviewed by: Dirk Werner
Mathematics Subject Classification ID: 46B80
Mathematics Subject Classification ID: 46B85
Mathematics Subject Classification ID: 46B06
Mathematics Subject Classification ID: 46-02
zbMATH DE Number: 6386234
zbMATH Keywords: Banach spaces
zbMATH Keywords: Lipschitz maps
zbMATH Keywords: Lipschitz free space
zbMATH Keywords: Arens-Eells space
zbMATH Keywords: coarse embeddings
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: Ribe programme
MaRDI profile type: Publication
arXiv ID: 1207.2958
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The non-linear geometry of Banach spaces after Nigel Kalton (OpenAlex ID)
OpenAlex ID: W2963603116