實變函數論(一)

This course is 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 analytic tools developed in the first half of the 20th century. We will treat real analysis mainly in the first semester and functional analysis in the second. The purpose is to lay down a solid foundation for further usage in some other theoretical or applied areas. After taking the course, the students are expected to have the general idea on the modern ways to attack the analysis problems.

Advanced calculus, general mathematical maturity

無備註

H. L. Royden, Real analysis , Collier-Macmillan Company, c1968.

(1) Class attendance: 1/10 of the final grade, (2) Assigned homework problems: 3/10 of the final grade, (3) Three midterm exams : 3/5 of the final grade.

• Preliminaries

  Points and Sets in R^n Metric space Open and closed sets Compact set Heine-Borel Theorem

• Preliminaries

  Functions Continuous functions and transformations The Riemann Integral

• Functions of bounded variations

  Function of bounded variation Rectifiable curves

• Functions of bounded variations

  The Riemann-Stieltjes integral More results about the Riemann-Stieltjes integral

• Lebesque and outer measure

  Lebesque outer measure Cantor set Lebesque measurable sets

• Lebesque and outer measure

  Two properties of Lebesque measure Characterizations of measurability

• Lebesque and outer measure

  Lipschitz transformations of R^n A nonmeasurable set

• Lebesque measurable functions

  Elementary properties of measurable functions Semicontinuous functions

• Lebesque measurable functions

  Egorov's Theorem and Luskin's Theorem Convergence in measure

• The Lebesque integral

  Definition of integral of a non-negative function properties of the integral

• The Lebesque integral

  The integral of an arbitrary measurable function The L^p space

• The Lebesque integral

  Riemann and Lebesque integrals

• Repeated integral

  Fibini's Theorem Tonelli's Theorem

• Repeated integral

  Applications of Fubini's Theorem

• Differentiation

  The indefinite integral Lebesque's differentiation Theorem

• Differentiation

  The Vitali Covering Lemma

• Differentiation

  Differentiation of Monotone functions Absolutely continuous and singular function

• Differentiation

  Absolutely continuous and singular function Convex function

Measure and integral: An introduction to real analysis Richard L Wheeden, Antoni Zygmund Taylor and Francis group, 1997.

地點

SA348

時間

Thursday 16:00~17:00

聯絡方式

Tel:56439