Department of Mathematics

Functional Analysis I

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Start of the lecture: Wednesday, 17.9.2014
Start of the exercise classes: Monday, 22.9.2014

Prof. Manfred Einsiedler
Coordinator Rene Rühr


Baire category, Banach spaces and linear maps.

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


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
Robin Krom
Monday 9-10
HG F 26.5
 Q-Z Maria Colombo Monday 9-10 HG G 26.1


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


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.
dual of l^p, corollaries of Hahn-Banach.
Chapter 7
Weak Topologies and Tychonoff-Alaoglu. Locally convex spaces and Frechet spaces.


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


Information to the exam.


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