實變函數論(二)

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: 2/10 of the final grade, (2) Assigned homework problems: 8/10 of the final grade,

• Differtiation

  Lebesque's differentiation Theorem Differentiation of monotone function

  備註：Materials in Unit 1~ Unit 11 are from the book "Measure and Integral".

• Differentiation

  Absolutely continuous and singular function Convex function

• L^p Class

  Definition of L^p Holder's inequality and Minkowski's inequality Classes l^p

• L^p Class

  Banach and metric spaces properties The space of L^2 Orthogonality

• L^p Class

  Fourier series Parseval's Formula Hilbert spaces

• Approximation of the identity; Maximal functions

  Convolution Approximation of the identity

• Approximation of the identity; Maximal functions

  Hardy-Littlewood Maximal functions The Marcinkiewicz integral

• Abstract Integration

  Additive set functions Measures

• Abstract Integration

  Measurable functions Integration

• Abstract Integration

  Absolutely continuous and singular set Functions and measures

• Abstract Integration

  The dual space of L^p Relative differentiation of measures

• Sobolev space

  Holder spaces Sobolev spaces

  備註：From now on, the materials are from Chapter 5 in Evan's book

• Sobolev space

  Approximation

• Sobolev space

  Extension Trace

• Sobolev space

  Sobolev inequalities Compactness

• Sobolev space

  Poincare inequality Difference quotient Differentiability Fourier transform methods

• Sobolev space

  The space H^{-1} Space involving time

Measure and integral: An introduction to real analysis Richard L Wheeden, Antoni Zygmund Taylor and Francis group, 1997. Partial Differential Equations Lawrence C. Evans Graduate studies in Mathematics AMS 1998

地點

SA348

時間

Thursday 16:00~17:00

聯絡方式

Tel:56439