Guirao Antonio J. et al. Open Problems in the Geometry and Analysis of Banach Spaces

Springer, 2016. — 179 p.This is a commented collection of some easily formulated open problems in the geometry and analysis on Banach spaces, focusing on basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology, and nonlinear geometry.The collection consists of some commented questions that, to our best knowledge, are open. In some cases, we associate the problem with the person we first learned it from. We apologize if it may turn out that this person was not the original source. If we took the problem from a recent book, instead of referring to the author of the problem, we sometimes refer to that bibliographic source. We apologize that some problems might have already been solved. Some of the problems are long-standing open problems, some are recent, some are more important, and some are only “local” problems. Some would require new ideas, and some may go only with a subtle combination of known facts. All of them document the need for further research in this area. The list has of course been influenced by our limited knowledge of such a large field. The text bears no intentions to be systematic or exhaustive. In fact, big parts of important areas are missing: for example, many results in local theory of spaces (i.e., structures of finite-dimensional subspaces), more results in Haar measures and their relatives, etc. With each problem, we tried to provide some information where more on the particular problem can be found. We hope that the list may help in showing borders of the present knowledge in some parts of Banach space theory and thus be of some assistance in preparing MSc and PhD theses in this area. We are sure that the readers will have no difficulty to consider as well problems related to the ones presented here. We believe that this survey can especially help researchers that are outside the centers of Banach space theory. We have tried to choose such open problems that may attract readers’ attention to areas surrounding them.Summing up, 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 a young researcher to choose his/her favorite part to work in.