Elements of Geometry of Balls in Banach Spaces

The passage from finite-dimensional vector spaces to infinite-dimensional ones, particularly in the historical context of quantum mechanics, presents significant foundational, algebraic, and geometric challenges that have fueled much of the development of modern mathematics. A most fruitful approach is embodied in the Banach space concept, namely a complete normed vector space (over the real or complex numbers). The norm endows each such space with a metric, and thus a rich geometry, linked with the algebraic structure through the Banach space axioms. Traditionally, one's first steps into mathematics take place in the finite-dimensional spaces \( \mathbb{R}^n \) or \(\mathbb{C}^n \) where both geometry and algebra are, from the perspective of Banach space theory, pretty simple. As soon as one ventures to infinite dimensions unexpected phenomena are encountered mandating a significant honing of one's intuition. The book under review offers a delightful perspective on the geometric aspects, leading the reader to marvel at the shapes and properties of balls in Banach spaces. 

 

Most of the book is concerned with infinite-dimensional spaces but the tour starts in the land of finite dimensions, as low as 2, 3, and 4. Already there the reader will find 'circles' of various shapes and several beautiful results (e.g., bounds on the length of the unit circle in dimension 2) which, while rather elementary, are not often encountered in the beginner's literature. Before continuing with the contents, a word about the style of the book is in order. The book is aimed at newcomers to functional analysis, requiring not much more than a level of familiarity with Banach spaces expected from a first course, either at the advanced undergraduate or early postgraduate level. Important and relevant theorems are introduced as the text progresses, frequently with references to the literature. The presentation itself assumes a certain level of maturity. Concepts are introduced quickly and the reader is trusted to engage herself with the concepts at the immediate level without being told to do so through exercises. Even though there are no prescribed exercises, the text is laden with nearly endless routes of investigation that the reader can and should pursue. This kind of approach is somewhat unusual among texts on the subject and I find the book to be very well pitched for the target audience. 

 

The book comprises about 160 pages in 11 chapters, resulting in short and condensed chapters, nearly all including a notes and comments section. I would describe the book as being splendidly dense - a masterfully planned guided tour that means business. Chapters 2-5 revolve around the theme of convexity and smoothness, measures of convexity and smoothness, their interplay, and, naturally, geometric features they exhibit. This main theme is the backbone of the rest of the book with the ensuing chapters addressing projections, convex sets, various geometric coefficients, radii and diameters, antipodal points, and more, exemplifying the effects of the main theme's concepts with these geometric constructs. 

 

The book thus addresses aspects of Banach space theory that are usually not treated in introductory books not for reasons of accessibility but rather for the necessity to introduce the cannons of the subject. The book is therefore a valuable companion to those making their first steps into Banach territory, offering an enrichment of the usual approach, pointing at side-ways off the main path that lead to beautiful sights and a heightened appreciation of the general theory. To the more advanced student or the expert the book is sure to harbor many interesting observations and insights. Occasionally, the authors naturally introduce concepts from elementary metric space theory which are not usual tenants in introductory courses. For instance, Chapter 10 on measures of noncompactness discusses the Kuratowski and Hausdorff measures of noncompactness. This is always done with geometry in mind and, other than providing ample opportunity for exercises, such concepts are immediately put to work in order to further understand Banach spaces. As always, smoothness and/or convexity find their way to the discussion. 

 

The book is suitable for self-study either alongside a traditional functional analysis book or as a standalone text for anybody with some knowledge of Banach spaces who is interested in an emphasis on geometric aspects. The book will also serve well as supplementary reading for standard courses. Alternatively, a one-semester course can easily be tailored around the book in various formats. The plethora of geometric ideas presented in the book and the particular style of the book also make the book suitable for project studies for ambitious students. 

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Ittay Weiss is a Senior Lecturer in Mathematics at the University of Portsmouth, UK