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Manuels CON (Browse shelf) Available 08774-01

Chapter headings: I. Hilbert spaces, II. Operators on Hilbert space, III. Banach spaces, IV. Locally convex spaces, V. Weak topologies, VI. Linear operators on a Banach space, VII. Banach algebras and spectral theory for operators on a Banach space, VIII. C∗-algebras, IX. Normal operators on Hilbert space, X. Unbounded operators, XI. Fredholm theory.
This book is an excellent text for a first graduate course in functional analysis. The early chapters are devoted to the various classes of commonly occurring topological vector spaces (with the exception of Chapter II). The exposition proceeds from the particular to the general, reflecting the author's stated belief that this develops the student's intuition better than the more austere approach of Bourbaki. Many interesting and important applications are included. For example, Banach limits and Runge's theorem are in Chapter III. In Chapter V the author develops the Stone-Čech compactification via Banach space duality, the Stone-Weierstrass theorem via the Kreĭn-Milʹman theorem (à la de Branges), and the existence of Haar measure on a compact group via the Ryll-Nardzewski fixed point theorem. In the reviewer's opinion, the author has chosen exactly the right topics on general topological vector spaces for a first course. For example, strict inductive limits and distributions are discussed, but the Mackey-Arens theorem, barrelled spaces, and the like, are not.
Operator theory makes an early appearance in the second chapter, where the spectral theorem and functional calculus for compact normal operators are established. As an application of the spectral theorem, the basic facts about Sturm-Liouville problems are obtained.
The second half of the book deals with operator theory in earnest. Chapter VI includes the Banach-Stone theorem and Lomonosov's theorem. Chapters VII and VIII are fairly standard. Chapter IX obtains the spectral theorem for bounded normal operators from the theory of abelian C∗-algebras and includes a treatment of multiplicity theory. The spectral theorem is generalized to the unbounded case in Chapter X, which also includes Stone's theorem and the Plancherel theorem. The final Chapter XI presents the basic facts about Fredholm and semi-Fredholm operators.
This book is a fine piece of work. It includes an abundance of exercises, and is written in the engaging and lucid style which we have come to expect from the author. (MathSciNet)

Bibliogr. p. [390]-393. Index

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