Stage 4 Analysis

Analysis:
Materials listed in stage 3, which have not been covered there, may be studied here.

Functional Analysis:

• Conway J.B. A Course in Functional Analysis
• Eidelman Y., Milman V. and Tsolomitis A. Functional Analysis: An Introduction
• Garrett P. Functional Analysis (FREE!)
• Hunter J. and Nachtergaele B. Applied Analysis or here (FREE!)
• Kreyszig E. Introductory Functional Analysis with Applications
• Riesz F. and Sz.-Nagy B. Functional Analysis - Riesz is a key figure in this field.
• Rudin W. Functional Analysis
• Schaefer H.H. with Wolff M.P. Topological Vector Spaces
• Serfaty S. Functional Analysis Notes [.pdf] (FREE!)
• Shilov G.E. Elementary Functional Analysis - Republication of Mathematical Analysis Vol. II, MIT Press, Cambridge, Mass.
• Teschl G. Functional Analysis (FREE!)
• Treves F. Topological Vector Spaces, Distributions and Kernels
• Ward T.B. Functional Analysis Lecture Notes [.pdf] (FREE!)
• Wilde I.F. C*-algebras (FREE!)
• Wilde I.F. Functional Analysis (FREE!)
  Wilde I.F. Functional Analysis (topological vector space version) (FREE!)
• Yosida K. Functional Analysis

Measure Theory:
See also the probability section below.

• Bogachev V.I. Measure Theory
• Cohn D.L. Measure Theory
• Fremlin D.H. Measure Theory (FREE!)
• Halmos P.R. Measure Theory - Another nice text by Halmos.
• Hjorth G. Measure Theory (FREE!)
• Liskevich V. Measure Theory [.pdf] (FREE!) - Elementary notes.
• Oxtoby J.C. Measure and Category: A Survey of the Analogies between Topological and Measure Spaces
• Sattinger D.H. Measure Theory & Integration [.pdf] (FREE!)
• Schilling R.L. Measures, Integrals and Martingales - A friendly introduction to these topics. Additional material, hints and solutions here.

Hilbert Spaces:

• Halmos P.R. A Hilbert Space Problem Book
• Halmos P.R. Introduction to Hilbert Space And the Theory of Spectral Multiplicity
• Young N. An Introduction to Hilbert Space

Real and Complex Analysis:

• Ahlfors L. Complex Analysis - Classic in complex analysis.
• Antimirov M.Y., Kolyshkin A.A. and Vaillancourt R. Complex Variables (here [.pdf] for FREE!)
• Ash R.B. and Novinger W.P. Complex Variables (here for FREE!)
• Baggett L. Analysis of Functions of a Single Variable (FREE!)
• Bartle R.G. A Modern Theory of Integration
• Bear H.S. A Primer of Lebesgue Integration
• Bergman S. The Kernel Function and Conformal Mapping
• Conway J.B. Functions of One Complex Variable I, II
• Duistermaat J.J. and Kolk J.A.C. Multidimensional Real Analysis I, II
• Folland G.B. Real Analysis: Modern Techniques and Their Applications
• Freitag E. and Busam R. Complex Analysis
• Hahn L.-S. and Epstein B. Classical Complex Analysis
• Krantz S.G. Function Theory of Several Complex Variables
• Krantz S.G. Partial Differential Equations and Complex Analysis
• Kurtz D.S. and Swartz C.W. Theories Of Integration: The Integrals Of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane
• Lang S. Real Analysis
• Markushevich A.I. Theory of Functions of a Complex Variable
• Needham T. Visual Complex Analysis - Plenty of diagrams, as the title suggested, make this book beautiful. See the web for extra information.
• Pedersen G.K. Analysis Now
• Royden H.L. Real Analysis - Fairly good text.
• Rudin W. Real and Complex Analysis - Big Rudin!
• Scheidemann V. Introduction to Complex Analysis in Several Variables
• Semmes S. Elements of Linear and Real Analysis (FREE!)
• Stein E.M. and Shakarchi R. Complex Analysis
• Stein E.M. and Shakarchi R. Real Analysis: Measure Theory, Integration, and Hilbert Spaces
• Sternberg S. Theory of Functions of Real Variable [.pdf] (FREE!)
• Whitney H. Geometric Integration Theory

Fourier and Harmonic Analysis:

• Axler S., Bourdon P. and Ramey W. Harmonic Function Theory (here for FREE!)
• Brown J.W. and Churchill R.V. Fourier Series and Boundary Value Problems
• Dahlberg B.E.J. and Kenig C.E. Harmonic Analysis and Partial Differential Equations [.ps] (FREE!)
• Dales H.G., Aiena P. Eschmeier J. Laursen K. and Willis G.A. Introduction to Banach Algebras, Operators, and Harmonic Analysis
• Hardy G.H. and Rogosinski W.W. Fourier Series
• Hewitt E. and Ross K.A. Abstract Harmonic Analysis: Volume 1: Structure of Topological Groups. Integration Theory. Group Representations
  Hewitt E. and Ross K.A. Abstract Harmonic Analysis: Volume 2: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups
• Howell K.B. Principles of Fourier Analysis
• Katznelson Y. An Introduction to Harmonic Analysis
• Semmes S. Adventures in Harmonic Analysis (FREE!)
• Stein E.M. and Shakarchi R. Fourier Analysis: An Introduction
• Stein E.M. Harmonic Analysis
• Tao T. Fourier Analysis 1, 2 (FREE!)
• Varadhan S.R.S. Harmonic Analysis (FREE!)
• Wolff T. Lectures on Harmonic Analysis (FREE!)
• 國立交通大學：傅立業分析與應用課程講義 (FREE!)