Análisis Funcional26679
- Centro
- Facultad de Ciencia y Tecnología
- Titulación
- Grado en Matemáticas
- Curso académico
- 2024/25
- Curso
- 4
- Nº Créditos
- 6
- Idiomas
- Inglés
- Código
- 26679
DocenciaAlternar navegación
Guía docenteAlternar navegación
Descripción y Contextualización de la AsignaturaAlternar navegación
Esta asignatura solo se imparte en inglés
OBJECTIVES
The objectives of the course are the study of the main properties of bounded operators between Banach and Hilbert spaces, of the basic results associated to the different types of convergences in normed spaces and for the spectral theorem and its applications.
COURSE DESCRIPTION
The Functional Analysis is an important branch of Mathematics developed with the purpose to cover theoretical needs of Partial Differential Equations and Mathematical Analysis. The Functional Analysis is related to problems arising on Partial Differential Equations, Measure Theory and other branches of Mathematics.
We do not encourage to register in the course to students with less than a B2 english level. To take the course we recommend to have first taken the courses: Calculus I (1º), Calculus II (2º), Complex Analysis (2º), Linear Algebra and Geometry I (1º), Linear Algebra and Geometry II (2º), Differential Equations (3º), Measure and Integration (3º) and Partial Differential Equations (4º).
Competencias/ Resultados de aprendizaje de la asignaturaAlternar navegación
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.
Contenidos teórico-prácticosAlternar navegación
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.
MetodologíaAlternar navegación
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.
Sistemas de evaluaciónAlternar navegación
- Sistema de Evaluación Final
- Herramientas y porcentajes de calificación:
- Prueba escrita a desarrollar (%): 85
- Realización de prácticas (ejercicios, casos o problemas) (%): 15
Convocatoria Ordinaria: Orientaciones y RenunciaAlternar navegación
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.
Convocatoria Extraordinaria: Orientaciones y RenunciaAlternar navegación
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.
Materiales de uso obligatorioAlternar navegación
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
BibliografíaAlternar navegación
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
Direcciones web
http://www.ehu.eus/luis.escauriaza/
GruposAlternar navegación
61 Teórico (Inglés - Mañana)Mostrar/ocultar subpáginas
Semanas | Lunes | Martes | Miércoles | Jueves | Viernes |
---|---|---|---|---|---|
1-3 | 13:00-14:00 (1) | ||||
1-15 | 12:00-13:00 (2) | 10:30-11:30 (3) | |||
11-15 | 13:00-14:00 (4) |
61 Seminario-1 (Inglés - Mañana)Mostrar/ocultar subpáginas
Semanas | Lunes | Martes | Miércoles | Jueves | Viernes |
---|---|---|---|---|---|
4-15 | 13:00-14:00 (1) |
61 P. de Aula-1 (Inglés - Mañana)Mostrar/ocultar subpáginas
Semanas | Lunes | Martes | Miércoles | Jueves | Viernes |
---|---|---|---|---|---|
1-15 | 08:30-09:30 (1) | ||||
5-9 | 13:00-14:00 (2) |