Advanced Real Analysis systematically develops those concepts and
tools in real analysis that are vital to every mathematician,
whether pure or applied, aspiring or established. Along with a
companion volume Basic Real Analysis (available separately or
together as a Set via the Related Links nearby), these works
present a comprehensive treatment with a global view of the
subject, emphasizing the connections between real analysis and
other branches of mathematics.
Key topics and features of Advanced Real Analysis:
* Develops Fourier analysis and functional analysis with an eye
toward partial differential equations
* Includes chapters on Sturma "Liouville theory, compact
self-adjoint operators, Euclidean Fourier analysis, topological
vector spaces and distributions, compact and locally compact
groups, and aspects of partial differential equations
* Contains chapters about analysis on manifolds and foundations
of probability
* Proceeds from the particular to the general, often introducing
examples well before a theory that incorporates them
* Includes many examples and nearly two hundred problems, and a
separate 45-page section gives hints or complete solutions for most
of the problems
* Incorporates, in the text and especially in the problems,
material in which real analysis is used in algebra, in topology, in
complex analysis, in probability, in differential geometry, and in
applied mathematics of various kinds
Advanced Real Analysis requires of the reader a first course in
measure theory, including an introduction to the Fourier transform
and to Hilbert and Banach spaces. Some familiarity with complex
analysis is helpful for certain chapters. Thebook is suitable as a
text in graduate courses such as Fourier and functional analysis,
modern analysis, and partial differential equations. Because it
focuses on what every young mathematician needs to know about real
analysis, the book is ideal both as a course text and for
self-study, especially for graduate students preparing for
qualifying examinations. Its scope and approach will appeal to
instructors and professors in nearly all areas of pure mathematics,
as well as applied mathematicians working in analytic areas such as
statistics, mathematical physics, and differential equations.
Indeed, the clarity and breadth of Advanced Real Analysis make it a
welcome addition to the personal library of every
mathematician.
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