A Comprehensive Course in Analysis

A Comprehensive Course in Analysis by Poincaré 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.

Readership

Researchers (mathematicians and some applied mathematicians and physicists) using analysis, professors teaching analysis at the graduate level, graduate students who need any kind of analysis in their work.

Table of Contents

Contents for Part 1 (Real Analysis)

  • Preliminaries
  • Topological spaces
  • A first look at Hilbert spaces and Fourier series
  • Measure theory
  • Convexity and Banach spaces
  • Tempered distributions and the Fourier transform
  • Bonus chapter: Probability basics
  • Bonus chapter: Hausdorff measure and dimension
  • Bonus chapter: Inductive limits and ordinary distributions
  • Bibliography
  • Symbol index
  • Subject index
  • Author index
  • Index of capsule biographies

Contents for Part 2A (Basic Complex Analysis)

  • Preliminaries
  • The Cauchy integral theorem: Basics
  • Consequences of the Cauchy integral formula
  • Chains and the ultimate Cauchy integral theorem
  • More consequences of the CIT
  • Spaces of analytic functions
  • Fractional linear transformations
  • Conformal maps
  • Zeros of analytic functions and product formulae
  • Elliptic functions
  • Selected additional topics
  • Bibliography
  • Symbol index
  • Subject index
  • Author index
  • Index of capsule biographies

Contents for Part 2B (Advanced Complex Analysis)

  • Riemannian metrics and complex analysis
  • Some topics in analytic number theory
  • Ordinary differential equations in the complex domain
  • Asymptotic methods
  • Univalent functions and Loewner evolution
  • Nevanlinna theory
  • Bibliography
  • Symbol index
  • Subject index
  • Author index
  • Index of capsule biographies

Contents for Part 3 (Harmonic Analysis)

  • Preliminaries
  • Pointwise convergence almost everywhere
  • Harmonic and subharmonic functions
  • Bonus chapter: Phase space analysis
  • Hp spaces and boundary values of analytic functions on the unit disk
  • Bonus chapter: More inequalities
  • Bibliography
  • Symbol index
  • Subject index
  • Author index
  • Index of capsule biographies

Contents for Part 4 (Operator Theory)

  • Preliminaries
  • Operator basics
  • Compact operators, mainly on a Hilbert space
  • Orthogonal polynomials
  • The spectral theorem
  • Banach algebras
  • Bonus chapter: Unbounded self-adjoint operators
  • Bibliography
  • Symbol index
  • Subject index
  • Author index
  • Index of capsule biographies

http://www.ams.org/bookstore-getitem/item=simon-set

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