Functional Analysis
Exercises
Exercise sheets will be handed out on Wednesdays, on the following Monday there will be held a preliminary discussion for selected exercises (marked on the exercise sheet). On the following Wednesday they have to be handed in. In some of the exercise classes there will be additional exercises (multiple choice quiz) which have to be solved during the exercise class. Students should reasonably solve 70% of all these exercises in order to be qualified for the exam. In addition, each student has to hold once a preliminary discussion or present the solution of one of the exercises (5-10 minutes). The exercises and a schedule for the multiple choice quiz are available here.
References
- The lecture notes can be found here.
- "Funktionalanalysis", Harro Heuser, in German.
- "Funktionalanalysis", Dirk Werner, in German, could be found here.
- "Lecture notes" by Prof. Struwe, in German, could be found here.
- "Beginning Functional analysis", Karen Saxe, in English.
- "A Course in Functional analysis", John B. Conway, in English.
- There are many other english books, e.g. by, Rudin ("Functional Analysis") or by Folland.
Outline of the Lectures
Lecture
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Topic
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Week 1 (21.9.)
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Motivation, Normed Vector Spaces (1.1, 1.2, 1.3, 2.1)
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Week 2 (26.9., 28.9.)
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Banachspaces, Stone-Weierstrass (2.2, 2.3)
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Week 3 (3.10., 5.10.)
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Bounded operators and functionals, Arzela-Ascoli, Connection to Ordinary Differential Equations (2.3, 2.4, 1.4)
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Week 4 (10.10., 12.10.)
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Hilbert Spaces, Riesz Representation Theorem (2.5)
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Week 5 (17.10., 19.10.)
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Orthonormal Bases, Fourier Series (2.5, 3.2)
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Week 6 (24.10., 26.10.)
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Dirichlet and Fejér kernel, Sobolev Spaces on the Torus, Uniform Boundedness (3.2, 3.4, 5.1)
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Week 7 (31.10., 2.11.)
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Open Mapping Theorem and consequences (5.2)
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Week 8 (7.11., 9.11.)
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Dual Spaces: Hahn-Banach and consequences (6.1)
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Week 9 (14.11., 16.11.)
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Duals of L^1 and C(X) (Riesz) (6.2, 6.3)
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Week 10 (21.11., 23.11.)
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Dual of L^p and C(X) (Riesz Representation), weak and weak* convergence (6.2, 6.3)
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Week 11 (28.11., 30.11.)
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Tychonoff Alaoglu, Equidistribution, Topologies on B(X,Y) (7.1, 7.2, 7.3)
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Week 12 (5.12., 7.12.)
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Locally convex spaces, convex sets, Krein-Milman (7.4)
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Week 13 (12.12., 14.12.)
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Repetition, Sobolev spaces: harmonic functions (3)
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Week 14 (19.12., 21.12.)
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Repetition, Sobolev spaces: harmonic functions (3)
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