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.

This book provides a comprehensive presentation of recent approaches to and results about properties of various classes of functional spaces, such as Banach spaces, uniformly convex spaces, function spaces, and Banach algebras. Each of the 12 articles in this book gives a broad overview of current subjects and presents open problems. Each article includes an extensive bibliography. This book is dedicated to Professor Per. H. Enflo, who made significant contributions to functional analysis and operator theory.

Handbook of the Geometry of Banach Spaces

This book is about the subject of higher smoothness in separable real Banach spaces. It brings together several angles of view on polynomials, both in finite and infinite setting. Also a rather thorough and systematic view of the more recent results, and the authors work is given. The book revolves around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available? The subject of infinite dimensional real higher smoothness is treated here for the first time in full detail, therefore this book may also serve as a reference book.

In recent years there has been a surge of profound new developments in various aspects of analysis whose connecting thread is the use of Banach space methods. Indeed, many problems seemingly far from the classical geometry of Banach spaces have been solved using Banach space techniques. This volume contains papers by participants of the conference "Banach Spaces and their Applications in Analysis", held in May 2006 at Miami University in Oxford, Ohio, in honor of Nigel Kalton's 60th birthday. In addition to research articles contributed by participants, the volume includes invited expository articles by principal speakers of the conference, who are leaders in their areas. These articles present overviews of new developments in each of the conference's main areas of emphasis, namely nonlinear theory, isomorphic theory of Banach spaces including connections with combinatorics and set theory, algebraic and homological methods in Banach spaces, approximation theory and algorithms in Banach spaces. This volume also contains an expository article about the deep and broad mathematical work of Nigel Kalton, written by his long time collaborator, Gilles Godefroy. Godefroy's article, and in fact the entire volume, illustrates the power and versatility of applications of Banach space methods and underlying connections between seemingly distant areas of analysis.

In this book, non-spectral methods are presented and discussed that have been developed over the last two decades for the investigation of asymptotic behavior of operator semigroups. This concerns in particular Markov semigroups in L1-spaces, motivated by applications to probability theory and dynamical systems. Related results, historical notes, exercises, and open problems accompany each chapter.

The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem. Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes. The contents of this extensive and powerful toolbox have been mostly scattered around in research papers and lecture notes. Collecting this diverse body of material into a unified and accessible presentation fills a gap in the existing literature. The principal audience that we have in mind consists of researchers who need and use Analysis in Banach Spaces as a tool for studying problems in partial differential equations, harmonic analysis, and stochastic analysis. Self-contained and offering complete proofs, this work is accessible to graduate students and researchers with a background in functional analysis or related areas.

Banach spaces provide a framework for linear and nonlinear functional analysis, operator theory, abstract analysis, probability, optimization and other branches of mathematics. This book introduces the reader to linear functional analysis and to related parts of infinite-dimensional Banach space theory. Key Features: - Develops classical theory, including weak topologies, locally convex space, Schauder bases and compact operator theory - Covers Radon-Nikodým property, finite-dimensional spaces and local theory on tensor products - Contains sections on uniform homeomorphisms and non-linear theory, Rosenthal's L1 theorem, fixed points, and more - Includes information about further topics and directions of research and some open problems at the end of each chapter - Provides numerous exercises for practice The text is suitable for graduate courses or for independent study. Prerequisites include basic courses in calculus and linear. Researchers in functional analysis will also benefit for this book as it can serve as a reference book.

One of the subjects of functional analysis is classification of Banach spaces depending on various properties of the unit ball. The need of such considerations comes from a number of applications to problems of mathematical analysis. The list of subjects includes: differential calculus in normed spaces, approximation theory, weak topologies and reflexivity, general theory of convexity and convex functions, metric fixed point theory and others. The book presents basic facts from this field.