Theory of Functions of a Real Variable's Sternberg

IntroductionI have taught the beginning graduate course in real variables and functionalanalysis three times in the last fifive years, and this book is the result. Thecourse assumes that the student has seen the basics of real variable theory andpoint set topology. The elements of the topology of metrics spaces are presented(in the nature of a rapid review) in Chapter I.The course itself consists of two parts: 1) measure theory and integration,and 2) Hilbert space theory, especially the spectral theorem and its applications.In Chapter II I do the basics of Hilbert space theory, i.e. what I can dowithout measure theory or the Lebesgue integral. The hero here (and perhapsfor the fifirst half of the course) is the Riesz representation theorem. Includedis the spectral theorem for compact self-adjoint operators and applications ofthis theorem to elliptic partial difffferential equations. The pde material followsclosely the treatment by Bers and Schecter inPartial Difffferential EquationsbyBers, John and Schecter AMS (1964)Chapter III is a rapid presentation of the basics about the Fourier transform.Chapter IV is concerned with measure theory. The fifirst part follows Caratheodory’sclassical presentation. The second part dealing with Hausdorffff measure and dimension, Hutchinson’s theorem and fractals is taken in large part from the bookby Edgar,Measure theory, Topology, and Fractal GeometrySpringer (1991).This book contains many more details and beautiful examples and pictures.Chapter V is a standard treatment of the Lebesgue integral.Chapters VI, and VIII deal with abstract measure theory and integration.These chapters basically follow the treatment by Loomis in hisAbstract Harmonic Analysis.Chapter VII develops the theory of Wiener measure and Brownian motionfollowing a classical paper by Ed Nelson published in the Journal of Mathematical Physics in 1964. Then we study the idea of a generalized random processas introduced by Gelfand and Vilenkin, but from a point of view taught to usby Dan Stroock.The rest of the book is devoted to the spectral theorem. We present threeproofs of this theorem. The fifirst, which is currently the most popular, derivesthe theorem from the Gelfand representation theorem for Banach algebras. Thisis presented in Chapter IX (for bounded operators). In this chapter we againfollow Loomis rather closely.In Chapter X we extend the proof to unbounded operators, following Loomisand Reed and SimonMethods of Modern Mathematical Physics. Then we giveLorch’s proof of the spectral theorem from his bookSpectral Theory. This hasthe flflavor of complex analysis. The third proof due to Davies, presented at theend of Chapter XII replaces complex analysis by almost complex analysis.The remaining chapters can be considered as giving more specialized information about the spectral theorem and its applications. Chapter XI is devoted to one parameter semi-groups, and especially to Stone’s theorem aboutthe infifinitesimal generator of one parameter groups of unitary transformations.Chapter XII discusses some theorems which are of importance in applications of the spectral theorem to quantum mechanics and quantum chemistry. Chapter XIII is a brief introduction to the Lax-Phillips theory of scattering.