Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with topology and measure theory, in particular Lebesgue integration and \(L^p\) spaces).
Baire category; Banach and Hilbert spaces, bounded linear operators; basic principles: uniform boundedness, open mapping/closed graph theorem, Hahn-Banach theorem; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu theorem; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces.
Michael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2019/20.
Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.
Elias M. Stein, Rami Shakarchi. Functional analysis (Volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011.
Manfred Einsiedler, Thomas Ward. Functional Analysis, Spectral Theory, and Applications. Graduate Text in Mathematics 276. Springer, 2017.
Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
After each lecture, the recording is published here. For further information, check the following information about lecture recording.
You can access all the material concerning the lectures at the following polybox folder.
| Week | Contents |
|---|---|
| 19.09 / 22.09 | Normed vector spaces and their basic properties (Struwe 2.1). |
| 26.09 / 29.09 | Continuous linear maps between normed vector spaces (Struwe 2.2). Quotient spaces (Section 2.3). Hahn-Banach theorem, real and complex version (Struwe 4.1). |
| 03.10 / 06.10 | Baire category theorem (Struwe 1.3). Applications to functional analysis (Struwe 3.1-3.3). |
| 10.10 / 13.10 | Closed graph theorem (Struwe 3.3). Hilbert spaces (Struwe 2.4). Duality in Hilbert spaces (Struwe 4.3). |
| 17.10 / 20.10 | Strongly diverging Fourier series. \(L^p\) spaces as completions. |
| 24.10 / 27.10 | Fourier transform on \(L^2\) for \(2\pi\)-periodic functions. Definition of weak convergence (Struwe 4.6). |
| 31.10 / 03.11 | Weak and weak\(^*\) topologies (Struwe 4.6 and 5.3). Reflexivity and separability (Struwe 5.2 and 5.1). |
| 07.11 / 10.11 | Separation theorems (Struwe 4.5). Convex sets and the weak topology (Struwe 4.6). Reflexivity and weak compactness (lecture notes on polybox). |
| 14.11 / 17.11 | Milman-Pettis theorem (Brezis 3.7). Adjoint operators and duality (Struwe 6.1 and 6.2). Compact operators (Struwe 6.3). |
| 21.11 / 24.11 | Examples of compact operators and their applications (lecture notes on polybox). Compactness criteria: Arzelà-Ascoli theorem (with proof) and Frechet-Kolmogorov theorem (without proof) (Struwe 6.3). Applications of the compactness criteria (lecture notes on polybox). |
| 28.11 / 01.12 | Fredholm operators (lecture notes on polybox). Holomorphic families of operators (lecure notes on Polybox). |
| 05.12 / 08.12 | Spectral theory: generalities (Struwe 6.5). Spectral theory of compact operators (Struwe 6.7 and lecture notes on polybox). |
| 12.12 / 15.12 | Spectral theory of compact, self-adjoint operators and related examples. Courant-Fischer-Weyl theorem (Struwe 6.7 and lecture notes on polybox). Functional calculus (lecture notes on Polybox). |
| 19.12 / 22.12 | Functional calculus and related examples (lecture notes on Polybox). |
Please register and enroll for a teaching assistant in myStudies. The enrollment is needed to attend the exercise class and to hand in your homework.
| Assistant | Room |
|---|---|
| Ata Deniz Aydin | HG G 26.5 |
| Cosmin Manea | ML J 34.1 |
| Andrea Nützi | HG G 26.1 |
| Pieter-Bart Peters | ML F 40 |
Every Friday, a new problem set is uploaded here. You have seven days to solve the problems and hand in your solutions via the platform SAMUpTool (the precise deadline is the following Friday, no later than 8pm). Your work will be carefully graded and given back to you after a few days. During exercise classes on Monday some of the problems will be discussed.
| Assignment date | Due date | Problem set | Solution |
|---|---|---|---|
| Fri 23.09. | Fri 30.09. | Exercise sheet 1 | Solutions |
| Fri 30.09. | Fri 07.10. | Exercise sheet 2 | Solutions |
| Fri 07.10. | Fri 14.10. | Exercise sheet 3 | Solutions |
| Fri 14.10. | Fri 21.10. | Exercise sheet 4 | Solutions |
| Fri 21.10. | Fri 28.10. | Exercise sheet 5 | Solutions |
| Fri 28.10. | Fri 04.11. | Exercise sheet 6 | Solutions |
| Fri 04.11. | Fri 11.11. | Exercise sheet 7 | Solutions |
| Fri 11.11. | Fri 18.11. | Exercise sheet 8 | Solutions |
| Fri 18.11. | Fri 25.11. | Exercise sheet 9 | Solutions |
| Fri 25.11. | Fri 02.12. | Exercise sheet 10 | Solutions |
| Fri 02.12. | Fri 09.12. | Exercise sheet 11 | Solutions |
| Fri 09.12. | Fri 16.12. | Exercise sheet 12 | Solutions |