Poincare奖得主BarrySimon的《分析综合教程》是一套五卷本的经典教程，可以作为研究生阶段的分析学教科书。这套分析教程提供了很多额外的信息，包含数百道习题和大量注释，这些注释扩展了正文内容并提供了相关知识的重要历史背景。阐述的深度和广度使这套教程成为几乎所有经典分析领域的宝贵参考资料。
第4部分侧重于算子理论，尤其是Hilbert空间。中心主题是谱定理、迹类理论和Fredholm行列式，以及无界自伴算子的研究。此外还介绍了正交多项式理论和关于Banach代数的长章，包括交换和非交换Gel'fand-Naimark定理以及对一般局部紧致Abel群的Fourier分析。
本书可供专业研究人员（数学家、部分应用数学家和物理学家）、讲授研究生阶段分析课程的教师以及在工作和学习中需要任何分析学知识的研究生阅读参考。

Preface to the Series
Preface to Part 4
Chapter 1. Preliminaries
§1.1. Notation and Terminology
§1.2. Some Complex Analysis
§1.3. Some Linear Algebra
§1.4. Finite-Dimensional Eigenvalue Perturbation Theory
§1.5. Some Results from Real Analysis
Chapter 2. Operator Basics
§2.1. Topologies and §pecial Classes of Operators
§2.2. The Spectrum
§2.3. The Analytic Functional Calculus
§2.4. The Square Root Lemma and the Polar Decomposition
Chapter 3. Compact Operators, Mainly on a Hilbert §pace
§3.1. Compact Operator Basics
§3.2. The Hilbert-§chmidt Theorem
§3.3. The Riesz-Schauder Theorem
§3.4. Ringrose Structure Theorems
§3.5. Singular Values and the Canonical Decomposition
§3.6. The Trace and Trace Class
§3.7. Bonus Section: Trace Ideals
§3.8. Hilbert-Schmidt Operators
§3.9. Schur Bases and the Schur-Lalesco-Weyl Inequality
§3.10. Determinants and Fredholm Theory
§3.11. Operators with Continuous Integral Kernels
§3.12. Lidski's Theorem
§3.13. Bonus Section: Regularized Determinants
§3.14. Bonus Section: Weyl's Invariance Theorem
§3.15. Bonus Section: Fredholm Operators and Their Index
§3.16. Bonus Section: M. Riesz's Criterion
Chapter 4. Orthogonal Polynomials
§4.1. Orthogonal Polynomials on the Real Line and Favard's Theorem
§4.2. The Bochner-Brenke Theorem
§4.3. L2-and L∞-Variational Principles: Chebyshev Polynomials
§4.4. Orthogonal Polynomials on the Unit Circle: Verblunsky's and Szego's Theorems
Chapter 5. The Spectral Theorem
§5.1. Three Versions of the Spectral Theorem: Resolutions of the Identity, the Functional Calculus, and Spectral Measures
§5.2. Cyclic Vectors
§5.3. A Proof of the Spectral Theorem
§5.4. Bonus Section: Multiplicity Theory
§5.5. Bonus Section: The Spectral Theorem for UnitaryOperators
§5.6. Commuting Self-adjoint and Normal Operators
§5.7. Bonus Section: Other Proofs of the Spectral Theorem
§5.8. Rank-One Perturbations
§5.9. Trace Class and Hilbert-Schmidt Perturbations
Chapter 6. Banach Algebras
§6.1. Banach Algebra: Basics and Examples
§6.2. The Gel'fand Spectrum and Gel'fand Transform
§6.3. Symmetric Involutions
§6.4. Commutative Gel'fand-Naimark Theorem and the Spectral Theorem for Bounded Normal Operators
§6.5. Compactifications
§6.6. Almost Periodic Functions
§6.7. The GNS Construction and the Noncommutative Gel'fand-Naimark Theorem
§6.8. Bonus Section: Representations of Locally Compact Groups
§6.9. Bonus Section: Fourier Analysis on LCA Groups
§6.10. Bonus Section: Introduction to Function Algebras
§6.11. Bonus Section: The L1(R) Wiener and Ingham Tauberian Theorems
§6.12. The Prime Number Theorem via Tauberian Theorems
Chapter 7. Bonus Chapter: Unbounded Self-adjoint Operators
§7.1. Basic Definitions and the Fundamental Criterion for Self-adjointness
§7.2. The Spectral Theorem for Unbounded Operators
§7.3. Stone's Theorem
§7.4. von Neumann's Theory of Self-adjoint Extensions
§7.5. Quadratic Form Methods
§7.6. Pointwise Positivity and Semigroup Methods
§7.7. Self-adjointness and the Moment Problem
§7.8. Compact, Rank-One and Trace Class Perturbations
§7.9. The Birman-Schwinger Principle
Bibliography
Symbol Index
Subject Index
Author Index
Index of Capsule Biographies