Functional Analysis [MA3001]
Wintersemester 2014/15
Prof. Dr. Michael M. Wolf
| Dozent: | Prof. Dr. Michael M. Wolf | |
| Übungsleitung: | Michael Kech Alexander Müller-Hermes |
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| Mitwirkende: | ||
| Vorlesung: | Tuesday, 14:15 - 15:45, PH HS 2 and Friday, 14:15 - 15:45 PH HS 1 | Anmeldung |
| Zentralübung: | Anmeldung | |
| Tutorübungen: | T01 Monday, 16:15 - 17:45, MI 03.10.011, Michael Kech T02 Wednesday, 08:30 - 10:00, MI 03.10.011, Alexander Müller-Hermes T03 Friday, 10:15 - 11:45, MI 03.08.011, Alexander Müller-Hermes T04 Wednesday, 10:15 - 11:45, MI 02.04.011, Martin Idel T05 Monday, 12:15-13:45, MI 03.06.011, Michael Kech |
Anmeldung |
Final exam
- Repetition Exam will be on 30.03.2015
- Note the holiday course: http://www.ma.tum.de/Ferienkurse/WiSe1415/FA
- The correction of the repeat exame is completed!
- Klausureinsicht: 15.04.2015, 17:00 Uhr, Room 03.12.020A
| File | Date | Content | Comments |
|---|---|---|---|
| Lecture 1 | 07.10.14 | Hierarchy of spaces, topological spaces and notions | |
| Lecture 2 | 10.10.14 | Compactness, metrizable spaces, topologically equivalent metrics, completion | |
| Lecture 3 | 14.10.14 | Closedness vs. completeness, Baire category theorem, normed spaces, sequence spaces | |
| Lecture 4 | 17.10.14 | Function spaces, absolutely convergent series, quotient spaces | |
| Lecture 5 | 21.10.14 | Mazur-Ulam, separability of normed spaces, finite dim. normed spaces | |
| Lecture 6 | 25.10.14 | Operators, functionals, dual spaces, BLT theorem | |
| Lecture 7 | 28.10.14 | Some dual space isomorphisms, Zorn's Lemma, Hahn-Banach theorem (real version) | |
| Lecture 8 | 31.10.14 | Hahn-Banach (complex and norm-preserving version), HB consequences, a dual variational problem, Banach spaces duals and separability | |
| Lecture 9 | 04.11.14 | Reflexivity, Separating points from convex sets, closed hyperplanes and bounded functionals | |
| Lecture 10 | 07.11.14 | Geometric Hahn-Banach, dual description of closed convex hulls | |
| Lecture 11 | 11.11.14 | Open mapping theorem, inverse mapping theorem, closed graph theorem | |
| Lecture 12 | 14.11.14 | Banach-Steinhaus theorem, weak and weak-* topologies, weak convergence | |
| Lecture 13 | 18.11.14 | Closure, convergence and boundedness in weak topologies, polars, bipolar theorem | |
| Lecture 14 | 21.11.14 | Banach-Alaoglu theorem, characterization of w*-compact sets and w-compact unit balls, Krein-Milman theorem | |
| Lecture 15 | 25.11.14 | Sesquilinear forms, polarization identities, scalar products, parallelogram law, Hilbert spaces, unique closest element thm. | |
| Lecture 16 | 28.11.14 | Orthogonal complements, (anti-)unitaries, Wigner's thm., orthonormal bases, Bessel's inequality | |
| Lecture 17 | 02.12.14 | ONBs and their properties, isomorphisms to l_2, Riesz representation theorem, reflexivity of Hilbert spaces | |
| Lecture 18 | 05.12.14 | Radon-Nikodym theorem, Lax-Milgram theorem, Adjoint operators and their properties | |
| Lecture 19 | 09.12.14 | Properties of the adjoint, classes of operators, projections, orthogonal projections | |
| Lecture 20 | 12.12.14 | Properties of hermitian and normal operators, positive partial order, square root and modulus of positive operators | |
| Lecture 21 | 16.12.14 | Polar decomposition and some consequences, trace, compact operators, trace-class, Hilbert-Schmidt class | |
| Lecture 22 | 19.12.14 | Finite rank, trace-class, Hilbert-Schmidt class and compact operators as ideals in B(H), variance inequality, Hilbert-Schmidt Hilbert space | |
| Lecture 23 | 09.01.15 | Cauchy-Schwarz for operators, dual spaces of compact and trace-class, spectral notions (resolvent, point/cont./residual spec.) | |
| Lecture 24 | 13.01.15 | Spectra of multiplication operators, Neumann series, GL(X), analyticity of resolvent, for B(X) spectra are compact and non-empty | |
| Lecture 25 | 16.01.15 | Gelfand's spectral radius formula, numerical range, spectra of normal, self-adjoint, positive, unitary and projection operators | |
| Lecture 26 | 20.01.15 | Spectral theorem for normal compact operators, Fredholm alternative, singular value/Schmidt decomposition for compact operators | |
| Lecture 27 | 27.01.15 | Schatten classes, Operator topologies (uniform, strong, weak, ultraweak), Spectral measures and resolutions, functional calculus | Outlook |
| Lecture 28 | 30.01.15 | Hellinger-Toeplitz theorem, basic notions around unbounded operators, Cayley transform, Stone's theorem | Outlook |
Content (from Module description)
Banach and Hilbert spaces; bounded linear operators, open mapping theorem; spectral theory for compact selfadjoint operators; duality, Hahn-Banach theorems; weak and weak* convergence; brief introduction to unbounded operatorsLiterature
There are many good books and scripts. Here are some of them. Apart from Bollobas and Brokate they go considerably beyond what we can do in one semester.- Peter D. Lax: Functional Analysis (Wiley, 2002) [good two semester course]
- Gert K. Pedersen: Analysis Now (Springer, 1989) [very compact, very elegant, but quite advanced]
- John B. Conway: A Course in Functional Analysis
(Springer, 1990) [standard book on the subject; two semester course; structure differs from our course]
- Dirk Werner: Funktionalanalysis (Springer, 1995) [good two semester course; German]
- Bela Bollobas: Linear Analysis (Cambridge University Press, 1990) [well written; based on a course given in Cambridge; closest to our lecture course]
- Martin Brokate: Funktionalanalysis (Vorlesungsskript) [good German script based on the course taught in WS13/14 at TUM]
