實分析導論

本課程屬大學部與研究所程度的實分析課程,授課偏重於數學理論與分析方法的介紹,是高等微積分的一部分也是進入實變數函數論的基礎課程。Topics covered includes: Lebesgue measure theory, basic functional analysis, the theory of singular integrals. Elements of the theory of distributions (generalized functions).

Introduction to Analysis, Calculus, Linear algebra

In this course, we will cover the following material from the classical textbook "Real and Complex Analysis" by Walter Rudin. (Third Edition) Chapter 1. The concept of measurability (page 8) Simple functions (page 15) Elementary properties of measures (page 16) Integration of positive functions (page 19) Remark: We covered these four sections within the first week of class already. Since the focus of this course is about the application of Lebesuge integration theory in functional analysis and harmonic analysis, we will take the existence of Lebesgue measure in the Euclidean space setting for granted. As such, in this course, we will not discuss the standard method for the construction of Lebesgue measure. As such, we will skip through Chapter 2 of Rudin textbook. All we do is to mention the notion and existence of positive Borel measure in the setting of locally compact Hausdoff space. All we need here is the conclusion of the Riesz Representation theorem as stated on page 40 of Rudin textbook. Our standpoint is just to accept results such as the Riesz Representation theorem for the existence of positive Borel measure, without going through the proofs of these results in details. However, our course will seriously cover the following material in great details: Chapter 3 Lp spaces. Convex functions and inequalities (page 61) The L^p spaces (page 65) Approximation by continuous functions (page 69) Chapter 4 Elementary Hilbert Space Theory Inner products and linear functionals (page 76) Orthonormal sets (page 82) Trigonometric series (page 88) Chapter 5 Examples of Banach Space Techniques Banach Spaces (page 95) Consequences of Baire's Theorem (page 97) Fourier Series of continuous functions (page 100) Fourier coefficients of L^1 functions (page 103) The Hahn-Banach theorem (page 104) After having such a solid background and understanding of these chapters and sections in the textbook of Rudin, we turn to study the first two chapters of the textbook "Singular integrals and Differentiability properties of functions" by E. Stein. However, the homework and the exams will mainly cover the above mentioned chapters and sections from the textbook "Real and Complex Analysis" by Rudin.

Homework 30% Notes taking 10% Mid Term 1 20% Mid Term 2 20% Final Exam 20%

"Real and Complex Analysis" by Walter Rudin. (Third Edition)