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Functional Analysis [MA3001]

Wintersemester 2015/16

Prof. Dr. Michael M. Wolf

Dozent: Prof. Dr. Michael M. Wolf
Übungsleitung:  
Mitwirkende:  
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 operators

File 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 Pfeil
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  

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.