| 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 |
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| 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 |
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| Lecture 5 |
27.10.15 |
Lp-spaces, spaces of differentiable functions, absolut convergence, Mazur-Ulam theorem |
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| Lecture 6 |
30.10.15 |
Finite dimensional and separable normed spaces |
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| Lecture 7 |
03.11.15 |
Operators, functionals, operator norm, dual spaces, boundedness=continuity |
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| Lecture 8 |
06.11.15 |
BLT theorem, isometrically isomorphic spaces, Zorn's Lemma, Hahn-Banach extension theorem |
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| Lecture 9 |
10.11.15 |
Hahn-Banach extension theorems (complex and normed versions) with applications, separable dual spaces |
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| Lecture 10 |
13.11.15 |
Reflexivity, algebraic interior points, separating points from convex sets, closed hyperplanes and bounded functionals |
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| Lecture 11 |
17.11.15 |
Geometric Hahn-Banach theorems and (counter-)examples, dual characterisation of closed convex hulls |
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| Lecture 12 |
20.11.15 |
Open mapping theorem, inverse mapping theorem with applications, closed graph theorem |
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| Lecture 13 |
24.11.15 |
Banach-Steinhaus/principle of uniform boundedness, weak and weak-* topology |
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| Lecture 14 |
27.11.15 |
Weak and weak-* convergence, weak and weak-* open sets |
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| Lecture 15 |
01.12.15 |
Boundedness of weakly convergent sequences, weakly closed convex sets, Banach-Alaoglu theorem |
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| Lecture 16 |
04.12.15 |
Faces and extreme points of convex sets, Krein-Milman theorem |
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| Lecture 17 |
08.12.15 |
Inner product-and Hilbert spaces, Polarization, orthogonal complements |
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| Lecture 18 |
11.12.15 |
Orthogonal decomposition, orthonormal bases, separable Hilbert spaces |
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| Lecture 19 |
15.12.15 |
Basis expansion and Parseval identity, examples, Riesz representation thm. and its consequences |
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| Lecture 20 |
18.12.15 |
Adjoint operators and their properties, properties of projections and orthogonal projections |
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| Lecture 21 |
22.12.15 |
Hermitian and normal operators, decomposition into Hermitian and anti-Hermitian part, positive partial order |
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| Lecture 22 |
08.01.16 |
Positive square roots and absolute values of operators, partial isometries, polar decomposition, decomposition into positive parts |
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| Lecture 23 |
12.01.16 |
Trace, Hilbert-Schmidt, Trace-class and Compact operators and their approximation via finite rank operators |
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| Lecture 24 |
15.01.16 |
Ideals in B(H), Hilbert-Schmidt Hilbert space, duals of the spaces of compact and trace-class operators |
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| Last few lectures |
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Please see the lecture notes from last year at http://www-m5.ma.tum.de/Allgemeines/MA3001_2014W |
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Lectures 27 and 28, marked as Outlook, are not covered in the exam. |
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For the spectral mapping theorem, see Theorem 8.1 at http://www.mth.kcl.ac.uk/~jerdos/OpTh/w8.pdf |
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