Open Problems in the Geometry and Analysis of Banach Spaces [electronic resource] /
Guirao, Antonio J.author.
Montesinos, Vicente.author.
Zizler, Václav.author.
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.
Includes bibliographical references and indexes.
Preface -- Basic linear structure -- Basic linear geometry -- Biorthogonal systems -- Smoothness, smooth approximation -- Nonlinear geometry -- Some more nonseparable problems -- Some applications -- Bibliography -- List of concepts and problems -- Symbol index -- Subject index. .
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.
Functional analysis.
Approximation theory.
Measure theory.
Convex geometry .
Discrete geometry.
Algebraic topology.
Functional Analysis.
Approximations and Expansions.
Measure and Integration.
Convex and Discrete Geometry.
Algebraic Topology.