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. Clicking on the Part x hyperlinks will load tentative tables of contents (for some volumes, indices are not yet in place). If you are thinking of using these books as a primary course text in AY 2015-16, feel free to email Barry Simon for timing, contacts who have used them, etc.  A facebook page for these books can be found here.  

New This can now be ordered (A.M.S. member price: $280 for the set of all five volumes).  Official "expected" publication date is Dec. 21, 2015.

New Sample sections, two per part are now posted.

Part 1 - Real Analysis
 Part 1 Real AnalysisPoint 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.

Selected topics include Bernstein Polynomials, Pointwise Converge of Fourier Series, Lp spaces, Brownian Motion, Measures on Polish Spaces, Haar Measure, Convexity, Alexandroff–Hausdorff and Banach-Mazur theorems, Knaster–Kuratowski Fan, Krein Milman Theorem, moment problems, fixed point theorems including existence of invariant subspaces for compact operators, Hermite expansions, Nyquist-Shannon sampling theorem, Riesz products, fundamental solutions of classical linear PDEs and the Malgrange-Ehrenpreis theorem, law of large numbers, central limit theorem, law of the iterated logarithm, Poisson processes, Markov chains, Carathéodory construction, inductive limits and ordinary distributions.
Part 2a - Basic Complex Analysis 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.

Selected topics include Goursat Argument, Ultimate and Ultra Cauchy Integral Formulas, Runge’s Theorem, complex interpolation, Marty’s Theorem, continued fraction analysis of real numbers, Riemann mapping theorem, Uniformization theorem (modulo results from Part 3), Mittag Leffler and Weirstrass product theorems, finite order and Hadamard product formula, Gamma function, Euler-Maclaurin Series and Stirling’s formula to all orders, Jensen’s formula and Blaschke products, Weierstrass and Jacobi elliptic functions, Jacobi theta functions, Paley-Wiener theorems, Hartog’s phenomenon, Poincaré’s theorem that in higher complex dimensions, the ball and polydisk are not conformally equivalent.
Part 2b - Advanced Complex Analysis 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.

Selected topics include Poincaré metric, Ahlfors-Robinson proof of Picard’s theorem, Bergmann kernel, Painlevé’s conformal mapping theorem, Jacobi 2- and 4-squares theorems, Dirichlet series, Dirichlet’s prime progression theorem, zeta function, prime number theorem, hypergeometric, Bessel and Airy functions,  Hankel and Sommerfeld contours, Laplace’s method, stationary phase, steepest descent, WKB, Koebe function, Loewner evolution and introduction to SLE, Nevanlinna’s First and Second Main theorems.
Part 3 - Harmonic Analysis Part 3 Harmonic Analysis. Maximal functions and pointwise limits, harmonic functions and potential theory, phase space analysis, Hp spaces, more inequalities.

Selected topics include Hardy-Littlewood maximal function, von Neumann and Birkhoff ergodic theorems, Weyl equidistribution, ergodicity of Gauss (continued fraction) map, ergodicity of geodesic flow on certain Riemann surfaces, Kingman subadditive ergodic theorem, Ruelle–Oseledec theorem, martingale convergence theorem, subharmonic functions, Perron’s method, spherical harmonics, Frostman’s theorem, Kellogg-Evans theorem, potential theory on Riemann surfaces, pseudo-differential operators, coherent states, wavelets, BMO, real interpolation and Marcinkiewicz theorem, Hardy-Littlewood-Sobolev inequalities, Sobolev spaces, Calderón-Zygmund method, Calderón-Vaillancourt estimates, Hypercontractive and Log-Sobolev estimates, Lieb Thirring  and CLR bounds, Tomas-Stein theorem.
Part 4 - Operator Theory Part 4 Operator Theory. Eigenvalue Perturbation Theory, Operator Basics, Compact Operators, Orthogonal Polynomials, Spectral Theory, Banach Algebras, Unbounded Self-Adjoint Operators.

Selected topics include analytic functional calculus, polar decomposition, Hilbert-Schmidt and Riesz-Schauder theorems, Ringrose structure theorems, trace ideals, trace and determinant, Lidskii’s theorem, index theory for Fredholm operators, OPRL, OPUC, Bochner-Brenke theorem, Chebyshev polnomials, spectral measures, spectral multiplicity theory, trace class perturbations and Krein spectral shift, Gel’fand transform, Gel’fand-Naimark theorems, almost periodic functions, Gel’fand-Raikov and Peter-Weyl theorems, Fourier analysis on LCA groups, Wiener and Ingham tauberian theorems and the prime number theorem, Spectral and Stone’s theorem for unbounded self-adjoint operators, von Neumann theory of self-adjoint extensions, quadratic forms, Birman-Krein-Vershik theory of self adjoint extensions, Kato’s inequality, Beurling-Deny theorems, moment problems, Birman-Schwinger principle.

NewHere are the final covers for the book!

Part 1. Real Analysis
Real Analysis

Part 2a. Basic Complex Analysis
Basic Complex Analysis

Part 2b. Advanced Complex Analysis
Advanced Complex Analysis

Part 3. Harmonic Analysis
Harmonic Analysis

Part 4. Operator Theory
Operator Theory