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Open problems in the geometry and analysis of Banach spaces. (English) Zbl 1351.46001

Cham: Springer (ISBN 978-3-319-33571-1/hbk; 978-3-319-33572-8/ebook). xii, 169 p. (2016).
In every field of mathematics there are monographs describing the present knowledge in that area. It is likewise important to have compendia telling us what we do not yet know; the book under review does exactly this for the theory of Banach spaces.
The authors have selected 304 open problems (actually, it seems that one or two have already been solved); for each of them they present information on partial solutions and pointers to the literature. The problems are grouped into seven chapters (Basic Linear Structure; Basic Linear Geometry; Biorthogonal Systems; Differentiability and Structure, Renormings; Nonlinear Geometry; Some More Nonseparable Problems; Some Applications) and they range from deep classical problems such as the separable quotient problem (Does every Banach space admit a separable quotient space?)to rather concrete questions that still have defied all attempts to answer them, e.g., Does every \(2\)-dimensional Hilbert space have Lindenstrauss’s Property (B)?
At the end of the book there are an extended and very useful subject index and an alphabetical list of concepts and problems that help the reader to locate problems involving a particular topic.
This collection will be attractive to researchers in Banach space theory and to prospective PhD students in this field.

MSC:

46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
00A27 Lists of open problems
46B03 Isomorphic theory (including renorming) of Banach spaces
46B04 Isometric theory of Banach spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B20 Geometry and structure of normed linear spaces
46B26 Nonseparable Banach spaces
46B80 Nonlinear classification of Banach spaces; nonlinear quotients
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