[前へ戻る]
   

 授業科目
 Course Title
解析学特論
Topics in Analysis 
 担当者
 Instructor
講師   瀬戸 道生  前学期 月曜日4時限
 単 位
 Credit
2

関連するディプロマポリシー Related Diploma Policy
時代の課題と社会の要請に応えた専門的知識と技能/Expert knowledge and skills to address the issues of the age and the demands of society
 
到達目標 Target to be Reached
After the course, the students are expected to be able to: •
understand the basic concept of measure theory,
describe the construction of Lebesgue integral,
use the famous three convergence theorems and Fubini's theorem,
understand the basic structure of L^p spaces and Radon-Nikodym theorem.

 
授業内容 Course Content
Theory of Lebesgue integral is crucial to understand modern analysis (functional analysis, probability theory etc.).
The pourpose of this lecture is to introduce its basic theory.
The course covers the following topics:
abstract measure and integration theory,
the Lebesgue measure and the Lebesgue integral on the Euclidean space,
convergence theorems, Fubini's theorem, L^p spaces and Radon-Nikodym theorem.
 
授業計画 Course Planning
01. Introduction
02. Measure theory 1 (sigma algebras and Borel sets)
03. Measure theory 2 (basic properties)
04. Construction of Lebesgue integral
05. Basic properties of Lebesgue integral
06. Monotone convergence Theorem
07. Lebesgue dominated convergence theorem
08. Fubini's theorem
09. L^p spaces 1 (Hoelder's inequality and Minkowski's inequality)
10. L^p spaces 2 (completeness)
11. Riesz representation theorem
12. Radon-Nikodym theorem 1 (absolute continuity)
13. Radon-Nikodym theorem 2 (proof)
14. Summary

Homework will be given in ``almost every" lecture.
Your solutions to the assigned homework will be graded and returned to you.
 
授業運営 Course Management
Lecture

 
評価方法 Evaluation Method
The final grade will be based on the homework.
 
オフィスアワー Office Hour (s)
The students may ask questions by e-mail.
 

参考書 Book (s) for Reference
W. Rudin,Real and Complex Analysis,3rd edition,McGraw-Hill Book Co.
荷見守助『関数解析入門』第2版[内田老鶴圃]

 
 
 
[前へ戻る]