ETH :: D-MATH :: Functional Analysis

Basic principles of functional analysis: Open mapping theorem, Closed graph theorem, Stone-Weierstrass Theorem, Hahn-Banach Theorem, Convexity, reflexive spaces

Spectral Theory: Sobolev spaces and elliptic regularity

Lecture

Monday 10 - 12	HG D 7.2
Wednesday 8 - 10	HG G 3

Exercise Classes

Begins second semester week.

A-D	Jonas Lührman	Monday 9-10	HG E 21
E-K	Berit Singer	Monday 9-10	HG F 26.3
L-P	Robin Krom	Monday 9-10	HG F 26.5
Q-Z	Maria Colombo	Monday 9-10	HG G 26.1

Series

Sheets can be dropped off at J68.

Lecture History

(Caveat: This is not supposed to be a complete list for exam relevant topics)


Chapter 1	Motivation: Differential Equation Exercises: Equidistribution, Integral equations and Integral operators, Green function, ∆ Boundary value problem
Chapter 2	Norms, Banach Spaces, Bounded Operators, Banach Algebras Continuous Functions on compact spaces (Arzela Ascoli and Stone Weierstrass), Continuous functions are dense in L^p Hilbert-Spaces basics, Isometries, Convex sets, ONB, Measure decomposition Exercises: lp spaces, compact subsets of {C_0, lp, c_0}, Continuous functions on locally compact spaces, Hardy spaces, examples of linear operators, smooth functions are dense in Lp, Restricting the Identity map, von Neumann Theorem, Bergman Kernel, No Projection from l^\infty to c_0, Shift operators. L^p is sometimes uniformly convex.
Chapter 3	Fourier Series, Differentiability vs Absolute Convergence, Sobolev spaces. Exercises: Equivalent formulations of Sobolev Spaces. Possible singularities of Sobolev functions. Boundary Problems on the Intervall. Partial differentiability and continuity. Decomposition for SO(2)-eigenfunctions.
Chapter 4	Banach space of Compact operators, Spectral Theorem for compact, self-adjoint operators. Integral Operators are compact. Exercises: First Instance of Functional Calculus (square root operator). Spectrum of the Graph Laplacian. Study of a special Integral Operator.
Chapter 5	Banach-Steinhaus Theorem, Baire Category, Open Mapping Theorem and Corollaries Exercises: More Applications of Banach-Steinhaus and Open Mapping Theorem (complemented spaces, quantitative solvability)
Chapter 6	Hahn-Bahn, Duals of c_0, l^1. Reflexive spaces and weak star convergence. Exercises: dual of l^p, corollaries of Hahn-Banach.
Chapter 7	Weak Topologies and Tychonoff-Alaoglu. Locally convex spaces and Frechet spaces.

Script

The lecture will be accompanied by the Lecture Notes on Functional Analysis by M. Einsiedler and T. Ward.

Exam

Information to the exam.

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More information

Serie 1	Solution 1
Serie 2	Solution 2
Serie 3	Solution 3
Serie 4	Solution 4
Serie 5	Solution 5
Serie 6	Solution 6
Serie 7	Solution 7
Serie 8	Solution 8
Serie 9	Solution 9
Serie 10	Solution 10
Serie 11	Solution 11
Serie 12	Solution 12
Serie 13	Solution 13 (update 1a)