Principles of Analysis: Measure, Integration, Functional Analysis, and Applications

This book is intended to provide a detailed and rigorous treatment of the essential aspects of measure theory, integration and functional analysis for graduate students of mathematics. It has an unusually broad scope for such a text and probably has enough material for a three or four-semester course.

The text begins with a discussion of topological and algebraic preliminaries. After that, there are three major parts: measure and integration, functional analysis and applications. The author assumes that the reader has a strong background in undergraduate real analysis and a basic knowledge of linear algebra. Elementary point set and metric space topology are treated in the preliminary material, but it would be useful if the reader has some exposure to that already. A few parts of the book use results from complex analysis such as Cauchy’s integral theorem.

The treatment of measure theory and integration is fairly standard. The Lebesgue integral is developed in Euclidean space, and the theory of Lp spaces and differentiation are also handled more or less conventionally. These are supplemented with an optional chapter that treats measures on locally compact spaces. Fourier analysis in Euclidean space is also included as an optional topic.

The chapters on functional analysis could constitute a whole course by themselves. The core includes treatments of Banach and Hilbert spaces with their standard theorems, locally convex spaces and weak topologies on normed spaces. Operator theory is here too, up to the spectral theorem for compact normal operators, but the author also treats Hilbert-Schmidt and trace class operators. Commutative Banach algebras are developed sufficiently to prove the Gelfand representation theorem. Then there is a collection of miscellaneous topics like the Krein-Milman theorem and its applications tucked in as a separate chapter.

The book has an encyclopedic feel to it as if the author wanted to include all the topics he thought every future analyst needed to know. Certainly, there are portions that are intended more as reference or background material, but other parts seem to be topics that the author found particularly interesting or valuable. The chapters in the applications section fall into this category. They include the theory of distributions (aka generalized functions), analysis on locally compact groups, and analysis on semigroups. The author concludes with a chapter on probability theory that is effectively a short course up to and including stochastic processes, martingales, and stochastic integration.

Students who work through the all the material in the text would indeed have a very complete background in analysis.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.