Theory of functions of a real variable
Shlomo Sternberg
IntroductionI have taught the beginning graduate course in real variables and functional
analysis three times in the last fifive years, and this book is the result. The
course assumes that the student has seen the basics of real variable theory and
point 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 do
without measure theory or the Lebesgue integral. The hero here (and perhaps
for the fifirst half of the course) is the Riesz representation theorem. Included
is the spectral theorem for compact self-adjoint operators and applications of
this theorem to elliptic partial difffferential equations. The pde material follows
closely the treatment by Bers and Schecter inPartial Difffferential Equationsby
Bers, 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’s
classical presentation. The second part dealing with Hausdorffff measure and dimension, Hutchinson’s theorem and fractals is taken in large part from the book
by 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 motion
following a classical paper by Ed Nelson published in the Journal of Mathematical Physics in 1964. Then we study the idea of a generalized random process
as introduced by Gelfand and Vilenkin, but from a point of view taught to us
by Dan Stroock.
The rest of the book is devoted to the spectral theorem. We present three
proofs of this theorem. The fifirst, which is currently the most popular, derives
the theorem from the Gelfand representation theorem for Banach algebras. This
is presented in Chapter IX (for bounded operators). In this chapter we again
follow Loomis rather closely.
In Chapter X we extend the proof to unbounded operators, following Loomis
and Reed and SimonMethods of Modern Mathematical Physics. Then we give
Lorch’s proof of the spectral theorem from his bookSpectral Theory. This has
the flflavor of complex analysis. The third proof due to Davies, presented at the
end 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 about
the 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.
analysis three times in the last fifive years, and this book is the result. The
course assumes that the student has seen the basics of real variable theory and
point 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 do
without measure theory or the Lebesgue integral. The hero here (and perhaps
for the fifirst half of the course) is the Riesz representation theorem. Included
is the spectral theorem for compact self-adjoint operators and applications of
this theorem to elliptic partial difffferential equations. The pde material follows
closely the treatment by Bers and Schecter inPartial Difffferential Equationsby
Bers, 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’s
classical presentation. The second part dealing with Hausdorffff measure and dimension, Hutchinson’s theorem and fractals is taken in large part from the book
by 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 motion
following a classical paper by Ed Nelson published in the Journal of Mathematical Physics in 1964. Then we study the idea of a generalized random process
as introduced by Gelfand and Vilenkin, but from a point of view taught to us
by Dan Stroock.
The rest of the book is devoted to the spectral theorem. We present three
proofs of this theorem. The fifirst, which is currently the most popular, derives
the theorem from the Gelfand representation theorem for Banach algebras. This
is presented in Chapter IX (for bounded operators). In this chapter we again
follow Loomis rather closely.
In Chapter X we extend the proof to unbounded operators, following Loomis
and Reed and SimonMethods of Modern Mathematical Physics. Then we give
Lorch’s proof of the spectral theorem from his bookSpectral Theory. This has
the flflavor of complex analysis. The third proof due to Davies, presented at the
end 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 about
the 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.
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