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
