**Graduate Analysis**

**Instructor: **Guanghua Ji

**Grading: **be absed on homework and final exam

**Time and Venue: **6-8, MON; B819

**Prerequisites: **point set topology, real analysis, elementary functional analysis, group theory...

**References:**

1, 童裕孙, 泛函分析教程, 复旦大学出版, 2008.

2, John B. Conway, A Course in Functional Analysis, GTM 96.

3, R. J. Zimmer, Essential Results of Functional Analysis, 1990.

4, A. Deitmar, S. Echterhoff, Principles of Harmonic Analysis, Springer.

5, G. B. Folland, A Course in Abstract Harmonic Analysis, 2ed.

6, 黎景辉， 冯绪宁， 拓扑群引论， 科学出版社.

7, Rodney Coleman, Calculus on Normed Vector Spaces, Springer.

8, E. Kowalski, Spectral theory in Hilbert Spaces, online, ETH, Zurich, FS 09.

9, Loukas Grafakos, Classical Fourier Analysis, Springer, GTM 249.

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**1**.**Measure and Integration**

1.1. Measurable Functions and Integration

1.2. The Riesz Representation Theorem and Fubini's Theorem

1.3. Lp Spaces and the Riesz-Fischer Theorem

1.4. The Stone-Weierstrass Theorem

1.5. Vector-Valued Integrals

2.**Topological vector spacess**

2.1. Examples of spaces

2.2. Examples of operators

2.3. Operator topologies and groups of operators

2.4. Banch-Alaoglu theorem

3.**Locally Compact Groups**

3.1. Topological groups

3.2. Haar measure: existence and uniqueness

3.3. The modular functions

3.4. Homogenous spaces and Convolution

3.5. Representations of locally compact groups

4.**Fourier transforms and distributions theory**

4.1. Convolution and Approximate Identities

4.2. The Schwartz Class and the Fourier Transform

4.3. The Inverse Fourier Transform and Plancherel's formula

4.4. The Space of Tempered Distributions

4.5. Sobolev spaces

**5. Compact Operators and Compact Groups**

5.1. Compact operators and Hilbert-Schmidt operators

5.2. Spectral theorem for compact operators

5.3. Unitary representation and the Peter-Weyl theorem

5.4. Fourier Analysis on compact groups

5.5. Induced representations

**6. General spectral theory**

6.1. Spectrum of an operator

6.2. Spectral theorem for self-adjoint operators

6.3. Spectral measures and projection-valued measures

6.4. Gelfand's theory of commutative C*-algebras

6.5. Mean ergodic theorem

***7. Unbounded operators on Hilbert spaces**

7.1. Domain, graphs, adjoints, and spectrum

7.2. Criterion for self-adjointness

7.3. Spectral theory for unbounded operators

7.4. The spectral theorem

7.5. The Laplace operator

***8. Calculus on normed linear spaces**

8.1. Differentiation

8.2. Mean value theorems

8.3. Higher Derivatives and Differentials

8.4. Taylor Theorems and Applications

8.5. The Inverse and Implicit Mapping Theorems

Final Exam of Graduate Analysis

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Functional analysis is a broad mathematical area with strong connections to many domains within mathematics and physics. This book, based on a first-year graduate course taught by Robert J. Zimmer at the University of Chicago, is a complete, concise presentation of fundamental ideas and theorems of functional analysis. It introduces essential notions and results from many areas of mathematics to which functional analysis makes important contributions, and it demonstrates the unity of perspective and technique made possible by the functional analytic approach.

Zimmer provides an introductory chapter summarizing measure theory and the elementary theory of Banach and Hilbert spaces, followed by a discussion of various examples of topological vector spaces, seminorms defining them, and natural classes of linear operators. He then presents basic results for a wide range of topics: convexity and fixed point theorems, compact operators, compact groups and their representations, spectral theory of bounded operators, ergodic theory, commutative C*-algebras, Fourier transforms, Sobolev embedding theorems, distributions, and elliptic differential operators. In treating all of these topics, Zimmer's emphasis is not on the development of all related machinery or on encyclopedic coverage but rather on the direct, complete presentation of central theorems and the structural framework and examples needed to understand them. Sets of exercises are included at the end of each chapter.