Course Introduction

• Prerequisite：advanced calculus
• Recommended for： graduate students
• Introduction：

This is the course designed to acquaint the graduate students in the applied mathematics department with basic ideas and tools in modern analysis. This comprises the subjects of real analysis and functional analysis, the analysis developed in the 20th century and after. We will treat real analysis mainly in the first semester and functional analysis in the second. Hopefully, we can lay down a solid foundation for further usage in some other theoretical or applied area. After taking the course, the students are expected to have the general idea on the modern ways to attack the analysis problems.

• Syllabus：

1. Lebesgue measure and integral: Measurable functions, Littlewood’s three principles, Fatou’s lemma, monotone convergence theorem, bounded (dominated) convergence theorem.
2. Differentiation and integration: Function of bounded variation, absolute continuous function, singular function, fundamental theorem of calculus, Jensen ineguality.
3. Banach space：Lp spaces, Minkowski and Hölder inequalities, Riesz representation theorem.
4. (Optional) Topology: Baire category theory, Ascoli-Arzelá theorem, Stone-Weierstrass theorem.
5. General measure theory: Signed measure, Hahn decomposition theorem, Jordan decomposition Lebesgue Decomposition theorem, Radon-Nikodym theorem, Fubini theorem, Tonelli theorem.

• Reference：

1. H. L. Royden, Real Analysis, 3rd ed., 1989.
2. A. Friedman, Foundations of Modern Analysis.
3. R. L. Wheeden and A. Zygmund Measure and Integral.