Home Teaching

Foundations of Modern Analysis 2

Spring 2008 Reception Hours: Monday, 14:00 - 15:00, Room 320, Shreiber building

Syllabus:

1. Foundations of functional analysis in Banach spaces: finite and infinite dimensional normed linear spaces, linear transformations, the open-mapping theorem and the closed-graph theorem, application to PDEs, the Hahn-Banach theorem, conjugate spaces and reflexive spaces, adjoint operators, the conjugates of L^p and C[0,1].
2. Compact operators: the Fredholm-Riesz-Schauder theory, elements of spectral theory.
3. Hilbert spaces: the projection theorem, orthonormal sets.
4. Operators and spectral theory in a Hilbert space: self-adjoint operators, eigenvalue problems for differential and integral equations.

Literature:

Avner Friedman, "Foundations of Modern Analysis", Dover Publications, New York, 1982. book mirror readers: http://www.djvu.com/ http://www.lizardtech.com/

A. N. Kolmogorov, S. V. Fomin, "Introductory real analysis", Dover Publications, New York, 1970. book

Exercises:

4.1.1-4.1.6; 4.2.1 -4.2.7; 4.3.1-4.3.7; 4.4.1-4.4.9; 4.5.4, 4.5.5; 4.6.1-4.6.4, 4.6.6, 4.6.7; 4.8.1, 4.8.9-4.8.11; 4.10.1-4.10.14; 4.13.1-4.13.4; 5.1.1-5.1.8; 5.3.1-5.3.7, 5.3.9, 5.3.11; 6.1.1, 6.1.3; 6.2.1, 6.2.2; 6.4.1-6.4.3; 6.7.3, 6.7.4; 6.9.2-6.9.5

Hints

For exams (those problems were removed, which were not solved in the class):

4.1.1-4.1.6; 4.2.1 -4.2.7; 4.3.1-4.3.7; 4.4.1-4.4.9; 4.5.4, 4.5.5; 4.6.1-4.6.4, 4.6.6, 4.6.7; 4.8.1, 4.8.9-4.8.11; 4.10.1-4.10.14; 4.13.1-4.13.4; 5.1.1-5.1.8; 6.1.1, 6.1.3; 6.4.1-6.4.2; 6.7.3, 6.7.4; 6.9.2-6.9.5

Tests 2008: a, b