A Comprehensive Course in Analysis by Poincare Prize winner Barry
Simon is a five-volume set that can serve as a graduate-level
analysis textbook with a lot of additional bonus information,
including hundreds of problems and numerous notes that extend the
text and provide important historical background. Depth and breadth
of exposition make this set a valuable reference source for almost
all areas of classical analysis. Part 4 focuses on operator theory,
especially on a Hilbert space. Central topics are the spectral
theorem, the theory of trace class and Fredholm determinants, and
the study of unbounded self-adjoint operators. There is also an
introduction to the theory of orthogonal polynomials and a long
chapter on Banach algebras, including the commutative and
non-commutative Gel'fand-Naimark theorems and Fourier analysis on
general locally compact abelian groups.
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