## Functional Analysis [MA3001]

### Wintersemester 2015/16

### Prof. Dr. Michael M. Wolf

Dozent: |
Prof. Dr. Michael M. Wolf | |

Übungsleitung: |
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Mitwirkende: |
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Vorlesung: |
Tuesday, 14:15 - 15:45, PH HS 2 and Friday, 14:15 - 15:45 PH HS 1 | Anmeldung |

Tutorübungen: |
Anmeldung |

### News

* The exam results are ready, online for TUM students, sent by email to LMU students. * Discussion of the finite-dimensional case added in the solution of problem 4/a, Exam 2.### 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 operatorsFile | Date | Content | Comments |
---|---|---|---|

Lecture 1 | 13.10.15 | Hierarchy of spaces, topological spaces and topological notions | More details on point set topology in Hatcher's notes ^{} |

Lecture 2 | 16.10.15 | Convergence, compactness, topological vector spaces, metrizable spaces | |

Lecture 3 | 20.10.15 | Isometries, completion of metric spaces, closedness vs. completeness | Notes by Milan Mosonyi |

Lecture 4 | 23.10.15 | Baire category theorem, normed and Banach spaces, spaces of sequences and continuous functions | |

Lecture 5 | 27.10.15 | Lp-spaces, spaces of differentiable functions, absolut convergence, Mazur-Ulam theorem | |

Lecture 6 | 30.10.15 | Finite dimensional and separable normed spaces | |

Lecture 7 | 03.11.15 | Operators, functionals, operator norm, dual spaces, boundedness=continuity | |

Lecture 8 | 06.11.15 | BLT theorem, isometrically isomorphic spaces, Zorn's Lemma, Hahn-Banach extension theorem | |

Lecture 9 | 10.11.15 | Hahn-Banach extension theorems (complex and normed versions) with applications, separable dual spaces | |

Lecture 10 | 13.11.15 | Reflexivity, algebraic interior points, separating points from convex sets, closed hyperplanes and bounded functionals | |

Lecture 11 | 17.11.15 | Geometric Hahn-Banach theorems and (counter-)examples, dual characterisation of closed convex hulls | |

Lecture 12 | 20.11.15 | Open mapping theorem, inverse mapping theorem with applications, closed graph theorem | |

Lecture 13 | 24.11.15 | Banach-Steinhaus/principle of uniform boundedness, weak and weak-* topology | |

Lecture 14 | 27.11.15 | Weak and weak-* convergence, weak and weak-* open sets | |

Lecture 15 | 01.12.15 | Boundedness of weakly convergent sequences, weakly closed convex sets, Banach-Alaoglu theorem | |

Lecture 16 | 04.12.15 | Faces and extreme points of convex sets, Krein-Milman theorem | |

Lecture 17 | 08.12.15 | Inner product-and Hilbert spaces, Polarization, orthogonal complements | |

Lecture 18 | 11.12.15 | Orthogonal decomposition, orthonormal bases, separable Hilbert spaces | |

Lecture 19 | 15.12.15 | Basis expansion and Parseval identity, examples, Riesz representation thm. and its consequences | |

Lecture 20 | 18.12.15 | Adjoint operators and their properties, properties of projections and orthogonal projections | |

Lecture 21 | 22.12.15 | Hermitian and normal operators, decomposition into Hermitian and anti-Hermitian part, positive partial order | |

Lecture 22 | 08.01.16 | Positive square roots and absolute values of operators, partial isometries, polar decomposition, decomposition into positive parts | |

Lecture 23 | 12.01.16 | Trace, Hilbert-Schmidt, Trace-class and Compact operators and their approximation via finite rank operators | |

Lecture 24 | 15.01.16 | Ideals in B(H), Hilbert-Schmidt Hilbert space, duals of the spaces of compact and trace-class operators | |

Last few lectures | Please see the lecture notes from last year at http://www-m5.ma.tum.de/Allgemeines/MA3001_2014W | ||

Lectures 27 and 28, marked as Outlook, are not covered in the exam. | |||

For the spectral mapping theorem, see Theorem 8.1 at http://www.mth.kcl.ac.uk/~jerdos/OpTh/w8.pdf |

File | Date | Content | Comments |
---|---|---|---|

Exercise class 1 | Week 19 - 23 October | Topological spaces | Solutions, also for the Homework |

Exercise class 2 | Week 26 - 30 October | l_p spaces | Solutions, also for the Homework |

Exercise class 3 | Week 2 - 6 November | Function spaces, finite vs. infinite-dimensional spaces | Solutions, also for the Homework |

Exercise class 4 | Week 9 - 13 November | Operator norm, dual spaces | Solutions, also for the Homework |

Exercise class 5 | Week 16 - 20 November | Hahn-Banach theorem | Solutions, also for the Homework |

Exercise class 6 | Week 23 - 27 November | Frechet derivative | Solutions, also for the Homework |

Exercise class 7 | Week 30 November - 4 December | Closed graph theorem, uniform boundedness, weak topologies | Solutions, also for the Homework |

Exercise class 8 | Week 7 - 11 December | Weak topology, Krein-Milman theorem | Solutions, also for the Homework |

Exercise class 9 | Week 7 - 11 December | Inner product, Hilbert spaces | Solutions, also for the Homework |

Exercise class 10 | Week 11 - 15 January | Projections, operator norm | Solutions, also for the Homework |

Exercise class 11 | Week 18 - 22 January | Hilbert space operators | Solutions, also for the Homework |

Exercise class 12 | Week 25 - 29 January | Spectra of operators | Solutions, also for the Homework |

Exercise class 13 | Week 01 - 05 February | Spectra of operators 2 | Solutions |

### Homework

Submission of the Homework is voluntary. Each week there will be 2-3 Homework exercises, one of which will be corrected and marked, but it will be disclosed only after the submission deadline which exercise is the one that will be marked. Those who achieve at least 50% of the score on average over the whole semester get a Bonus: this means that they get an improvement of one step of their final exam grade. However, only those can benefit from the Bonus whose mark without the Bonus is sufficient for passing the exam. Homework has to be submitted to the tutor of your exercise class by the deadline specified on the exercise sheet each week. Homework has to be submitted individually; group submissions are not accepted.### Literature

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; good two semester 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]