A Comprehensive Course in Analysis by Barry Simon

In the second half of 2015, the American Math Society will publish a five volume (total about 3200 pages) set of books that is a graduate analysis text with lots of additional bonus material. Included are hundreds of problems and copious notes which extend the text and provide historical background.  Efforts have been made to find simple and elegant proofs and to keeping the writing style clear

Part 1 Real Analysis:

Point Set Topology, Banach and Hilbert Space, Measure Theory, Fourier Series and Transforms, Distribution Theory, Locally Convex Spaces, Basics of Probability Theory, Hausdorff Measure and Dimension.

Part 2a Basic Complex Analysis:

Cauchy Integral Theorem, Consequences of the Cauchy Integral Theorem (including holomorphic iff analytic, Local Behavior, Phragmén-Lindelöf, Reflection Principle, Calculation of Integrals), Montel, Vitali and Hurwitz’s Theorems, Fractional Linear Transformations, Conformal Maps, Zeros and Product Formulae, Elliptic Functions, Global Analytic Functions, Picard’s Theorem.

Part 2b Advanced Complex Analysis:

Conformal metric methods, topics in analytic number theory,  Fuchsian ODEs and associated special functions, asymptotic methods, univalent functions, Nevanlinna theory.

Part 3 Harmonic Analysis:

Maximal functions and pointwise limits, harmonic functions and potential theory, phase space analysis, Hp spaces, more inequalities.

Part 4 Operator Theory:

Eigenvalue Perturbation Theory, Operator Basics, Compact Operators, Orthogonal Polynomials, Spectral Theory, Banach Algebras, Unbounded Self-Adjoint Operators.

Reference: Click Here

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