ETH :: D-MATH :: Funktionalanalysis

Baire category; Banach spaces and linear operators; Fundamental theorems: Open Mapping Theorem, Closed Range Theorem, Uniform Boundedness Principle, Hahn-Banach Theorem; Convexity; reflexive spaces; Spectral theory.

Prerequisites

Students are supposed to have knowledge of real analysis. If you feel that you have gaps in this field, consider the following books:

"Real Analysis", Norman B. Haaser, Joseph A. Sullivan
"Functional Analysis", Frigyes Riesz, Bela Sz.-Nagy (part 1)

Exercises

Students should submit 60% of their home assignments in order to be qualified for the exam. The exercises are available here here.

References

"Lecture notes by Prof. Struwe". In German, could be found here.
"Funktionalanalysis", Dirk Werner. In German, could be found could be found here.
"Lecture notes by Prof. Salamon". In English. Could be found here.
There are many excellent textbooks in English.

Outline of the Lectures

Lecture	Date	Topic
Week 1 - Lecture 1.	22.9	Short lecture. Open, closed, etc.
Week 2 - Lecture 2.	27.9	Baire Category.
Week 2 - Lecture 3.	29.9	Applications
Week 3 - Lecture 4.	4.10	Normed spaces : up to p.17 in Struwe, definition of compactness.
Week 3 - Lecture 5.	6.10	Proof of compactness characterization and the result dim(unit ball) < infinity iff it is compact.
Week 4 - Lecture 6.	11.10	Ascoli Arzela and linear operators.
Week 4 - Lecture 7.	13.10	Linear operators ; end of chapter 2. in Struwe.
Week 5 - Lecture 8.	18.10	Main theorems of chapter 3.
Week 5 - Lecture 9.	20.10	Main theorems of chapter 3.
Week 6 - Lecture 10.	25.10	Main theorems of chapter 3.
Week 6 - Lecture 11.	27.10	Main theorems of chapter 3.
Week 7 - Lecture 12.	1.11	Hahn-Banach theorem.
Week 7 - Lecture 13.	3.11	Applications of Hahn-Banach theorem.
Week 8 - Lecture 14.	8.11	Dual spaces : pages 40-42 in Struwe.
Week 8 - Lecture 15.	10.11	Hilbert spaces : pages 42-46 in Struwe.
Week 9 - Lecture 16.	15.11	Dual of L^p spaces : pages 46-50 in Struwe.
Week 9 - Lecture 17.	17.11	Separation of convex sets : pages 50-53 in Struwe.
Week 10 - Lecture 18.	22.11	no class.
Week 10 - Lecture 19.	24.11	Weak covnergenceand convexity : pages 53-55 in Struwe.
Week 11 - Lecture 20.	29.11	Reflexive and separable spaces : pages 57-60 in Struwe.
Week 11 - Lecture 21.	1.12	Weak* convergnece ; pages 61-64 in Struwe.
Week 12 - Lecture 22.	6.12	Variational problems : pages 64-66 in Struwe.
Week 12 - Lecture 23.	8.12	Spectral theory : Chapter 6.
Week 13 - Lecture 24.	13.12	Spectral theory : Chapter 6.
Week 13 - Lecture 25.	15.12	Spectral theory : Chapter 6.
Week 14 - Lecture 26.	20.12	Spectral theory : Chapter 6.
Week 14 - Lecture 27.	22.12	Spectral theory : Chapter 6.

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