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**Course Code**: B12004Y

**Course Name**: Real Analysis

**Credits**: 4.0

**Level**: Undergraduate

**Pre-requisite**: Mathematical Analysis

**Lecture Time**: 80 periods

**Course Description**

This course is a required course for undergraduate students with major in mathematics and applied mathematics in University of Chinese Academy of Sciences. This course mainly presents the abstract integral theory (Lebesgue integral, etc.), measure, space theory, differential, integral in product space. After studying through this course, students are required to understand the basic concepts, methods, techniques and important theorems of the real analysis. It can lay a solid foundation for further study of modern mathematics and mathematics research work.

**Topics and Schedule **

1. Abstract Integration (12 periods)

1.1. Basic concepts of measurable functions, simple functions and integrable functions

1.2. Basic properties of measure

1.3. Convergence of function series

1.4. Lebesgue's monotone convergence theorem

1.5. Fatou lemma

1.6. Dominated convergence theorem

Teaching Focus: Integrable function series and its their convergence, Fatou’s lemma

Difficulty: Dominated convergence theorem

2. Positive Borel Measures (14 periods)

2.1. Basic concepts in topology

2.2. Riesz representation theorem

2.3. Regularity of Borel measure

2.4. Lebesgue measure

2.5. Continuity of measurable function

2.6. Lusin theorem

Teaching Focus: Riesz representation theorem, Lusin (Lu Jin) theorem

Difficulty: Riesz representation theorem, Regularity of Borel measure

3. Spaces (6 periods)

3.1. Convex function

3.2. Jensen inequality

3.3. Important inequalities in space: Holder inequality, Minkowski inequality

3.4. The relationship among several types of convergence in function series: norm convergence, convergence in measure, and almost sure convergence almost everywhere

Teaching Focus: Jensen inequality, Holder inequality, Minkowski inequality

Difficulty: The relationship among several types of convergence

4. Elementary Hilbert Space Theory (8 periods)

4.1. Scalar product

4.2. Parallelogram law

4.3.Projection theorem

4.4. Orthogonal basis

4.5. Fourier series

Teaching Focus: Projection theorem, Orthogonal basis

Difficulty: Fourier series

5. Examples of Banach Space Techniques (10 periods)

5.1. Normed space

5.2. Bell theorem and its corollary

5.3. Hahn-Banach theorem

5.4. Poisson integral

Teaching Focus: Hahn-Banach theorem

Difficulty: Baire theorem and its corollary

6.Complex Measures (12 periods)

6.1. Total variation

6.2. Absolute continuity

6.3. Radon-Nikodym theorem and its corollary

6.4. Bounded linear functional in space

Teaching Focus: Radon-Nikodym derivative, Bounded linear functional in space

Difficulty: Radon-Nikodym theorem and its corollary

7. Differentiation (8 periods)

7.1. Derivative of the measure

7.2. Hardy-Littlewood maximum function

7.3. Fundamental theorem of calculus

7.4. Differentiable transformation

Teaching Focus: Derivative of the measure, Fundamental theorem of calculus

Difficulty: Hardy-Littlewood maximum function

8. Integration on Product Spaces (6 periods)

8.1. Measure in product space

8.2. Fubini theorem

8.3. Completion of product measure

8.4. Convolution

8.5. Distribution function

Teaching Focus: Fubini theorem

Difficulty: Convolution, Distribution function

**Textbook**

Walter Rudin，Real and Complex Analysis, Third Edition, China Machine Press, 2006

**References**

Zhou Minqiang, Real Function Theory, Second Edition, Peking University Press, 2008