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An introduction to Banach space theory. (English) Zbl 0910.46008
Graduate Texts in Mathematics. 183. New York, NY: Springer. xix, 596 p. (1998).
This book is intended “for the student who has just completed a course on measure theory and integration that introduces the \(L_{p}\) spaces and would like to know more about Banach spaces in general” (from the blurb). It consists of five chapters entitled “Basic Concepts”, “The Weak and Weak\(^{*}\) Topologies”, “Linear Operators”, “Schauder Bases”, and “Rotundity and Smoothness”; in addition there are three appendices on metric spaces, the spaces \(\ell_{p}\), and ultranets, and a fourth one meticulously describes the prerequisites.
Chapter 1 develops basic notions of Functional Analysis in the context of Banach spaces such as the Hahn-Banach extension theorem, the open mapping theorem, the uniform boundedness principle and basic facts on reflexive spaces. Also, several characterisations of reflexive spaces due to R. C. James are proved, including his theorem on norm-attaining functionals. Chapter 2 starts with a detailed discussion of topological vector spaces and locally convex spaces, and eventually the weak, weak\(^{*}\) and bounded weak\(^{*}\) topologies of a Banach space and its dual are defined. Special emphasis is placed on characterisations of weakly compact sets culminating in a proof of James’ general weak compactness theorem. Also, the Krein-Milman and Bishop-Phelps theorems are proved. Chapter 3 presents a brief account of commutative Banach algebras and the spectral theory of compact operators. The notion of a weakly compact operator is defined as well, as is the Dunford-Pettis property. Chapter 4 deals with Schauder bases up to the Bessaga-Pełczyński selection principle, Pełczyński’s \(c_{0}\)-theorem and James’ theory of shrinking and boundedly complete unconditional bases. A final section introduces James’ famous quasi-reflexive space. In Chapter 5 the author considers strictly convex, uniformly convex and locally uniformly rotund spaces and spaces with Gâteaux, Fréchet and uniformly Fréchet differentiable norms and investigates various relations between these concepts.
The strength of this book lies in its very detailed approach; almost never does the author indulge in the popular “it is easy to see that …” (even if it is). Therefore students who do not want to travel light will probably like it. The author’s skill and precision of type-setting are impressive; I didn’t spot a single typo. Also, the wide collection of exercises and the references to original sources are welcome. The index is diligently arranged and comprehensive, and all arguments are complete and intelligible to the inexperienced reader. However, there is a price to pay for such a detailed exposition; remember that the book is nearly 600 pages long. For instance, in Chapter 2 it takes the author some 60 pages to prepare the definition of the weak topology, precious space part of which might have been filled with other material. I for one was disappointed that the author has neglected the past 25 years of Banach space theory almost entirely. For example, there is no mention of topics such as absolutely summing operators, Asplund spaces, Banach-Mazur distance, factorisation theorems (à la Davis-Figiel-Johnson-Pełczyński), finite representability and the principle of local reflexivity, \({\mathcal L}_{p}\)-spaces, super-properties and ultrapowers, Tsirelson-type spaces, type and cotype, etc. So, unfortunately this book doesn’t convey any flavour of modern Banach space theory but essentially sticks to the state of affairs as presented in Mahlon M. Day’s 1972 edition of his “Normed Linear Spaces”. In fact, Megginson’s book can be described as an introduction to Day’s terse treatise at an undergraduate / first year graduate (in American terms) level. On the other hand, his book doesn’t quite serve as an introduction to Functional Analysis at large, since Hilbert spaces and applications in other areas of analysis are (intentionally) not covered and not even mentioned. That more recent topics of Banach space theory along with an introduction to Functional Analysis can be treated in a fairly elementary fashion is splendidly proved by P. Habala, P. Hájek and V. Zizler in their “Introduction to Banach Spaces” (Matfyzpress, Prague 1996) which aims at an audience similar to Megginson’s. In conclusion, this text is ideal for readers seeking a detailed account of the most basic classical results in Banach space theory; topics developed within the last 25 or so years are not discussed here.
Reviewer: D.Werner (Berlin)

MSC:
46B10 Duality and reflexivity in normed linear and Banach spaces
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B20 Geometry and structure of normed linear spaces
47B07 Linear operators defined by compactness properties
46A04 Locally convex Fréchet spaces and (DF)-spaces
46B45 Banach sequence spaces
46A50 Compactness in topological linear spaces; angelic spaces, etc.
46J05 General theory of commutative topological algebras
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