History of Banach spaces and linear operators.

*(English)*Zbl 1121.46002
Basel: Birkhäuser (ISBN 978-0-8176-4367-6/hbk; 978-0-8176-4596-0/ebook). xxiii, 855 p. (2007).

This fundamental and interesting monograph is devoted to the history of a remarkable and rich branch of mathematics, the theory of (bounded) linear operators in Banach (complete normed linear) spaces. More than just a historical account, one can find here a comprehensive and deep treatment of basic concepts, methods, and results in this theory, the history of their appearance and development, and the interaction between this theory and the other branches of functional analysis and other parts of mathematics, such as set theory and mathematical logic, algebra and topology, differential and integral equations, probability theory and combinatorics, harmonic and numerical analysis, and so on. To read this book is not easy, but the material is highly intriguing and on almost every page the reader may discover something new or unexpected.

The book contains altogether eight chapters. The first four ones are not large and deal with the history of the theory in the first half of the 20th century. They form approximately a quarter of this book.

Chapter 1, “The Birth of Banach Spaces”, describes the history of the notion of Banach spaces. Here the author discusses notions of Minkowski (1896) and Hilbert (1906) spaces, Albert A. Bennet’s approach leading to complete normed Abelian groups (Newton’s method, 1913), and Kenneth W. Lamson’s approach leading to Banach spaces of scalar-valued functions on arbitrary sets (implicit function, 1920), the cumbersome system of Norbert Wiener’s axioms (1920), really leading to Banach spaces, and also approaches by Eduard Helly (1921, normed linear spaces of sequences) and by Hans Hahn (1921, spaces with duality).

Chapter 2, “Historical Roots and Basic Results”, contains sections on operators, functionals and dual operators, the moment problem and Hahn–Banach theorem, the uniform boundedness principle, the closed graph theorem and the open mapping theorem, and Riesz–Schauder theory. The chapter is completed with some remarks about Banach’s monograph (1931–1932) that defined the subsequent way for the development as the theory of linear operators in Banach spaces well as functional analysis in general.

Chapter 3, “Topological Concepts – Weak Topologies”, gathers parts devoted to problems connected with weak convergence and the weak topology in Banach spaces (weakly convergent sequences, topological spaces and topological linear spaces, locally convex linear spaces and duality, weak\(^*\) and weak compactness, weak sequential compactness and Schur property, transfinitely closed sets).

Chapter 4, “Classical Banach Spaces”, is a distinctive bridge between abstract Banach spaces and the concrete Banach spaces of classical analysis. The chapter consists of the following sections: Banach lattices, measures and integrals on abstract sets, the duality between \(L_1\) and \(L_\infty\), the Banach space \(L_p\), Banach spaces of continuous functions, measures and integrals on topological spaces, measures versus integrals, abstract \(L_p\)- and \(M\)-spaces, structure theory, operator ideals and operator algebras, complexification.

Chapters 5–7 contain the main body of this book. These chapters are concerned with many subjects and themes of functional analysis, and a significant part of these subjects and themes now became independent branches of functional analysis (or even mathematics in general). The author could find a successful variant for the presentation of this ocean of notations, methods, results, reasonings and arguments. Numerous diagrams make clear and transparent the relations between various properties of Banach spaces and linear operators acting between them, describe correlations and interactions between different classes of Banach spaces and linear operators, and allow the reader to observe the field as a whole.

Chapter 5, “Basic Results from the Post-Banach Period”, is concerned with the following parts of functional analysis: analysis in Banach spaces, spectral theory (including spectral operators), semigroups of operators, convexity, extremal points and related topics, geometry of the unit ball, bases (including wavelets), tensor products and approximation properties.

Chapter 6, “Modern Banach Space Theory – Selected Topics”, is devoted to the following parts of functional analysis: the geometry of Banach spaces (Banach–Mazur distance, Dvoretsky’s theorem, the history of Rademacher, Gauss, Fourier, Walsh, Haar type and cotype, volume ratios and Grothendieck numbers), \(s\)-numbers, operator ideals, eigenvalue distributions, traces and determinants, interpolation theory, function spaces (Hölder–Lipschitz, Sobolev, Besov, Lizorkin–Triebel spaces, Hardy spaces, Bergman spaces, Orlicz spaces), probability theory on Banach spaces, and further topics (topological properties of Banach spaces, stable Banach spaces, functors in categories of Banach spaces, the linear group of a Banach space, manifolds modelled on Banach spaces, asymptotic geometric analysis, and more).

Chapter 7, “Miscellaneous Topics”, is rather small, but it presents a highly readworthy discussion the following topics: Banach space theory as a part of mathematics, spaces versus operators, modern techniques of Banach space theory (probabilistic and combinatorial methods), counterexamples, Banach spaces and axiomatic set theory.

The final (and small) Chapter 8, “Mathematics Is Made by Mathematicians”, is divided into sections dealing with: victims of politics, scientific schools in Banach space theory, short biographies of some famous mathematicians (Frigyes Riesz, Eduard Helly, Stefan Banach, John von Neumann, Mark Grigorievich Krein, Alexander Grothendieck, Nicolas Bourbaki), Banach space theory at the ICMs, the Banach space archive, Banach space mathematicians, anniversary volumes and articles, obituaries.

At the end of the book, a chronology, original quotations, a bibliography (textbooks and monographs, historical and biographical books, collected and selected works, collections, seminars, anonymous works, mathematical papers, coauthors), and an index are given.

Of course, no book can cover the theory of linear operators in Banach spaces as a whole. Several aspects of Banach space geometry are not treated (such as Lumer’s product, numerous inequalities for elements of Banach spaces, angles and opening (gap) between subspaces, fractional powers of linear operators, Banach spaces of measurable functions, linear differential equations in Banach spaces, curve integrals and theory of primitives in Banach spaces, analogues of the Stone–Weierstrass theorem for continuous functions between Banach spaces, and others), but there is no need to reproach the author for these omissions. Rather, one must be thankful to him for this book that allows any mathematician to make sense of this rich and complicated field, viz. Banach spaces and linear operators acting between them.

There are two well-known books in the history of functional analysis [A. F. Monna, “Functional analysis in historical perspective” (Oosthoek, Utrecht) (1973; Zbl 0266.46001); J. Dieudonné, “History of functional analysis” (North–Holland Math. Studies 49) (1981; Zbl 0478.46001)]. The present book is essentially distinguished from those, regarding both its volume as well as its rich contents. For one thing, its basic part is devoted to the period after the 1950s of the 20th century. Moreover, it also treats many problems which have not yet been solved.

The author writes: “The book should be useful for readers who are interested in the question Why and how something happened.” I think, this is really true. I recommend this book to all who are interested in functional analysis, to beginners and researchers, all students and all professors who deal with functional analysis and its applications. Furthermore, I think that this book will be a most useful addition to any mathematical library.

The book contains altogether eight chapters. The first four ones are not large and deal with the history of the theory in the first half of the 20th century. They form approximately a quarter of this book.

Chapter 1, “The Birth of Banach Spaces”, describes the history of the notion of Banach spaces. Here the author discusses notions of Minkowski (1896) and Hilbert (1906) spaces, Albert A. Bennet’s approach leading to complete normed Abelian groups (Newton’s method, 1913), and Kenneth W. Lamson’s approach leading to Banach spaces of scalar-valued functions on arbitrary sets (implicit function, 1920), the cumbersome system of Norbert Wiener’s axioms (1920), really leading to Banach spaces, and also approaches by Eduard Helly (1921, normed linear spaces of sequences) and by Hans Hahn (1921, spaces with duality).

Chapter 2, “Historical Roots and Basic Results”, contains sections on operators, functionals and dual operators, the moment problem and Hahn–Banach theorem, the uniform boundedness principle, the closed graph theorem and the open mapping theorem, and Riesz–Schauder theory. The chapter is completed with some remarks about Banach’s monograph (1931–1932) that defined the subsequent way for the development as the theory of linear operators in Banach spaces well as functional analysis in general.

Chapter 3, “Topological Concepts – Weak Topologies”, gathers parts devoted to problems connected with weak convergence and the weak topology in Banach spaces (weakly convergent sequences, topological spaces and topological linear spaces, locally convex linear spaces and duality, weak\(^*\) and weak compactness, weak sequential compactness and Schur property, transfinitely closed sets).

Chapter 4, “Classical Banach Spaces”, is a distinctive bridge between abstract Banach spaces and the concrete Banach spaces of classical analysis. The chapter consists of the following sections: Banach lattices, measures and integrals on abstract sets, the duality between \(L_1\) and \(L_\infty\), the Banach space \(L_p\), Banach spaces of continuous functions, measures and integrals on topological spaces, measures versus integrals, abstract \(L_p\)- and \(M\)-spaces, structure theory, operator ideals and operator algebras, complexification.

Chapters 5–7 contain the main body of this book. These chapters are concerned with many subjects and themes of functional analysis, and a significant part of these subjects and themes now became independent branches of functional analysis (or even mathematics in general). The author could find a successful variant for the presentation of this ocean of notations, methods, results, reasonings and arguments. Numerous diagrams make clear and transparent the relations between various properties of Banach spaces and linear operators acting between them, describe correlations and interactions between different classes of Banach spaces and linear operators, and allow the reader to observe the field as a whole.

Chapter 5, “Basic Results from the Post-Banach Period”, is concerned with the following parts of functional analysis: analysis in Banach spaces, spectral theory (including spectral operators), semigroups of operators, convexity, extremal points and related topics, geometry of the unit ball, bases (including wavelets), tensor products and approximation properties.

Chapter 6, “Modern Banach Space Theory – Selected Topics”, is devoted to the following parts of functional analysis: the geometry of Banach spaces (Banach–Mazur distance, Dvoretsky’s theorem, the history of Rademacher, Gauss, Fourier, Walsh, Haar type and cotype, volume ratios and Grothendieck numbers), \(s\)-numbers, operator ideals, eigenvalue distributions, traces and determinants, interpolation theory, function spaces (Hölder–Lipschitz, Sobolev, Besov, Lizorkin–Triebel spaces, Hardy spaces, Bergman spaces, Orlicz spaces), probability theory on Banach spaces, and further topics (topological properties of Banach spaces, stable Banach spaces, functors in categories of Banach spaces, the linear group of a Banach space, manifolds modelled on Banach spaces, asymptotic geometric analysis, and more).

Chapter 7, “Miscellaneous Topics”, is rather small, but it presents a highly readworthy discussion the following topics: Banach space theory as a part of mathematics, spaces versus operators, modern techniques of Banach space theory (probabilistic and combinatorial methods), counterexamples, Banach spaces and axiomatic set theory.

The final (and small) Chapter 8, “Mathematics Is Made by Mathematicians”, is divided into sections dealing with: victims of politics, scientific schools in Banach space theory, short biographies of some famous mathematicians (Frigyes Riesz, Eduard Helly, Stefan Banach, John von Neumann, Mark Grigorievich Krein, Alexander Grothendieck, Nicolas Bourbaki), Banach space theory at the ICMs, the Banach space archive, Banach space mathematicians, anniversary volumes and articles, obituaries.

At the end of the book, a chronology, original quotations, a bibliography (textbooks and monographs, historical and biographical books, collected and selected works, collections, seminars, anonymous works, mathematical papers, coauthors), and an index are given.

Of course, no book can cover the theory of linear operators in Banach spaces as a whole. Several aspects of Banach space geometry are not treated (such as Lumer’s product, numerous inequalities for elements of Banach spaces, angles and opening (gap) between subspaces, fractional powers of linear operators, Banach spaces of measurable functions, linear differential equations in Banach spaces, curve integrals and theory of primitives in Banach spaces, analogues of the Stone–Weierstrass theorem for continuous functions between Banach spaces, and others), but there is no need to reproach the author for these omissions. Rather, one must be thankful to him for this book that allows any mathematician to make sense of this rich and complicated field, viz. Banach spaces and linear operators acting between them.

There are two well-known books in the history of functional analysis [A. F. Monna, “Functional analysis in historical perspective” (Oosthoek, Utrecht) (1973; Zbl 0266.46001); J. Dieudonné, “History of functional analysis” (North–Holland Math. Studies 49) (1981; Zbl 0478.46001)]. The present book is essentially distinguished from those, regarding both its volume as well as its rich contents. For one thing, its basic part is devoted to the period after the 1950s of the 20th century. Moreover, it also treats many problems which have not yet been solved.

The author writes: “The book should be useful for readers who are interested in the question Why and how something happened.” I think, this is really true. I recommend this book to all who are interested in functional analysis, to beginners and researchers, all students and all professors who deal with functional analysis and its applications. Furthermore, I think that this book will be a most useful addition to any mathematical library.

Reviewer: Peter Zabreiko (Minsk)