Springer Open Problems in the Geometry and Analysis of Banach Spaces

This is an collection of some easily-formulated problems that remain open in the study of the geometry and analysis of Banach spaces. Assuming the reader has a working familiarity with the basic results of Banach space theory, the authors focus on concepts of basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology and nonlinear geometry. The main purpose of this work is to help in convincing young researchers in Functional Analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems. Inside the Banach space area, the text should help expose young researchers to the depth and breadth of the work that remains, and to provide the perspective necessary to choose a direction for further study. Some of the problems are longstanding open problems, some are recent, some are more important and some are only local problems. Some would require new ideas, some may be resolved with only a subtle combination of known facts. Regardless of their origin or longevity, each of these problems documents the need for further research in this area. Review “In every field of mathematics there are monographs describing the present knowledge in that area. … the book under review does exactly this for the theory of Banach spaces. … 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. … attractive to researchers in Banach space theory and to prospective PhD students in this field.” (Dirk Werner, zbMATH 1351.46001, 2017) “The book is very well organized - every problem is preceded by an introductory part containing the notions and previous results necessary for its understanding, as well as references to significant papers or books containing partial solutions or related results. … All in all, the authors produced a marvelous piece of mathematical writing of great use for researchers in various fields of functional and mathematical analysis as well as for young graduate or PhD students.” (S. Cobzaş, Studia Universitatis Babes-Bolyai, Mathematica, Vol. 61 (4), 2016) From the Back Cover This is a collection of some easily-formulated problems that remain open in the study of the geometry and analysis of Banach spaces. Assuming the reader has a working familiarity with the basic results of Banach space theory, the authors focus on concepts of basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology and nonlinear geometry. The main purpose of this work is to help convince young researchers in Functional Analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems. Inside the Banach space area, the text should help expose young researchers to the depth and breadth of the work that remains, and to provide the perspective necessary to choose a direction for further study. Some of the problems presented herein are longstanding open problems, some are recent, some are more important and some are only "local" problems. Some would require new ideas, while others may be resolved with only a subtle combination of known facts. Regardless of their origin or longevity, each of these problems documents the need for further research in this area.