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Functional Analysis [MA3001]

Wintersemester 2014/15

Prof. Dr. Michael M. Wolf

Dozent: Prof. Dr. Michael M. Wolf
Übungsleitung: Michael Kech
Alexander Müller-Hermes
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

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 operators

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.