COMPETENCIES



CM04- To understand the concepts of Banach and Hilbert spaces and to learn to classify the standard examples. In particular, spaces of sequences and functions.

CM05- To learn to use properly the specific techniques for bounded operators over normed and Hilbert spaces.

CM06- To understand how to use the main properties of compact operators.

CM07- To learn to explain the fundamental results in the theory with accuracy and rigour.

CM08- To apply the spectral analysis of compact self-adjoint operators to the resolution of integral equations.



LEARNING OUTCOMES



To learn to recognize the fundamental properties of normed spaces and of the transformations between them. To be acquainted with the statement of the Hahn-Banach theorem and its corollaries. To understand the notions of dot product and Hilbert space. To apply the spectral theorem to the resolution of integral equations and Sturm-Liouville problems.

1. BANACH AND HILBERT SPACES: Banach spaces, finite dimensional normed spaces, examples of Banach spaces, Hilbert spaces, best approximation, projection theorem, dual of a Hilbert space, Riesz-Fréchet theorem, variational problems, the Dirichlet principle, bases in Hilbert spaces, orthogonality.

2. HAHN-BANACH THEOREM AND ITS CONSEQUENCES: Hahn-Banach theorem, the extension property. Topological dual of classical spaces. Weak topology and reflexive spaces.

3. SPECTRAL THEOREM: Spectral theorem for self-adjoint compact operators: examples of bounded operators on Hilbert spaces, inversion of operators, spectrum, adjoint of operators on a Hilbert space, compact operators, some applications of the spectral theorem.

4. BAIRE THEOREM AND ITS COROLLARIES: open mapping theorem, uniform boundedness theorem and closed graph theorem.

The standard ones: lectures, problem sessions and personal homeworks solved by the students with the help of the lecturers.



The theoretical contents will be presented in master classes following basic references in the bibliography. The lectures will be complemented with problem sessions, where the students will apply the theory explained in the lectures to solve some problem sets and to understand some of its applications. In the problem sessions, exercises and representative examples will be considered. These will given to the students in advance for them to have time to work out the solutions. Students must participate actively in the problem sessions. The discussion of its solutions will be encouraged.

Final written examination with questions related to the theory and problems worked out during the lectures. Students will turn in on the day of the final examination the written solutions to some of the problems assigned during the course.



Written examination: not less than 85% of the final grade.



Homework evaluation: not more than 15% of the final score.



The final grade will be No presentado when the written examination is not turned in.

In the lectures and problem sessions we shall mainly use the books:

K. Saxe. Beginning Functional Analysis. Springer
W. Rudin. Real and Complex Analysis. MacGrow-Hill Company.
H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer.
W. Rudin. Functional Analysis. McGraw-Hill Book Company.

and the hand written lecture notes in the web page

http://www.ehu.eus/luis.escauriaza/apuntes_problemas_y_examene/lecture-notes-functional.pdf

Bibliografía básica

The book K. Saxe. Beginning Functional Analysis. Springer together with the hand written lecture notes in the web page



http://www.ehu.eus/luis.escauriaza/apuntes_problemas_y_examene/lecture-notes-functional.pdf

Bibliografía de profundización

Yosida, K.: Functional Analysis, Springer-Verlag, 6th edition, 1980
Schechter, M.: Principles of Functional Analysis, AMS, 2nd edition, 2001
Hutson, V., Pym, J.S., Cloud M.J.: Applications of Functional Analysis and Operator Theory, 2nd edition, Elsevier Science, 2005, ISBN 0-444-51790-1
Dunford, N. and Schwartz, J.T. : Linear Operators, General Theory, and other 3 volumes, includes visualization charts
Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, AMS, 1963
Lebedev, L.P. and Vorovich, I.I.: Functional Anlysis in Mechanics, Springer-Verlag, 2002