實變函數論（二）

Abstract

課程首頁

本課程是由交通大學應用數學系提供。

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.

課程目標/概述 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. 課程章節

單元主題

內容綱要

Functional analysis in Banach spaces

Principle of uniform boundedness, open mapping theorem,closed graph theorem, Hahn-Banach theorem, completeness of Lp-spaces

Compact operators

Fredholm-Riesz-Schauder theory for compact operators

Spectral theory of self-adjoint operators Self-adjoint operators, normal operators

Hilbert spaces Basic properties of Hilbert spaces

課程書目

H. L. Royden, Real analysis, third edition,Macmillan, 1989. A. Torchinsky, Real variables, Addison-Wesley, 1988. J. B. Conway, A course in functional analysis, 2nd edition, Springer, 1990.

評分標準

項目 百分比

作業 1/3

期中考 1/3

期末考 1/3

授課對象：碩士班學生

預備知識：Advanced calculus and general mathematical maturity.