Functional Analysis

To understand and assimilate the fundamental principles of the Functional Analysis for Hilbert and normed spaces. To introduce some basic applications of Functional Analysis to the study of Differential Equations.

1. Hilbert spaces. (1h)
2. Normed spaces. Examples. (1h)
3. Operators. Adjoint of an operator. (2h)
4. Spectrum of an operator. (2h)
5. Theorem of Lax-Milgram. (2h)
6. Spectral decomposition of a compact and self-adjoint operator. (2h)
7. Integral equations. (2h)
8. Distributions. (2h)
9. Sobolev spaces. (2h)
10. Variational formulation of boundary value problems. (2h)

Basic bibliography:
L. Abellanas, A. Galindo, Espacios de Hilbert (Geometría, Operadores, Espectros), EUDEMA, 1987.

Complementary bibliography:
H. Brezis, Análisis Funcional, Alianza Universidad Textos, 1984.
B. Cascales, J.M. Mira, J. Origuela, y M. Raja, Análisis Funcional. Electrolibris : Real Sociedad Matemática Española, 2013.
J. Cerdá, Linear Functional Analysis, American Mathematical Society, 2010.

BASIC AND GENERAL COMPETENCES
GENERAL
CG01 - Introduce students into the research, as an integral part of a deep formation, preparing them for the eventual completion of a doctoral thesis.
CG02 - Acquisition of high level mathematical tools for diverse applications covering the expectations of graduates in mathematics and other basic sciences.
CG03 - Know the broad panorama of current mathematics, both in its lines of research, as well as in methodologies, resources and problems it addresses in various fields.
CG04 - Train for the analysis, formulation and resolution of problems in new or unfamiliar environments, within broader contexts.
CG05 - Prepare for decision making based on abstract considerations, to organize and plan and to solve complex issues.
BASICS
CB6 - Possess and understand knowledge that provides a basis or opportunity to be original in the development and / or application of ideas, often in a research context.
CB7 - That students know how to apply the knowledge acquired and their ability to solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their area of study.
CB8 - That students are able to integrate knowledge and face the complexity of making judgments based on information that, being incomplete or limited, includes reflections on social and ethical responsibilities linked to the application of their knowledge and judgments.
CB9 - That students know how to communicate their conclusions and the knowledge and ultimate reasons that sustain them to specialized and non-specialized audiences in a clear and unambiguous way.
CB10 - That students have the learning skills that allow them to continue studying in a way that will be largely self-directed or autonomous.
TRANSVERSAL COMPETENCES
CT01 - Use bibliography and search tools for general and specific bibliographic resources of Mathematics, including Internet access.
CT02 - Optimally manage work time and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision making.
CT03 - Enhance capacity for work in cooperative and multidisciplinary environments.
SPECIFIC COMPETENCES
CE01 - Train for the study and research in mathematical theories in development.
CE02 - Apply the tools of mathematics in various fields of science, technology and social sciences.
CE03 - Develop the necessary skills for the transmission of mathematics, oral and written, both in regard to formal correction, as well as in terms of communicative effectiveness, emphasizing the use of appropriate ICT.

To know the fundamentals elements and results of Functional Analysis. To use the methods of Functional Analysis in other areas of Mathematics.

The general methodological indications established in the Memory of the master’s degree in Mathematics of the University of Santiago de Compostela (USC) will be followed.

The teaching is programmed in expository and interactive classes.

Expository Teaching (9 hours): The lectures will be dedicated to the presentation and development of the essential contents of the subject.

Seminar and laboratory teaching (12 hours).

The general criteria of the Master Program.
Final Grade depends on the continuous evaluation (AC) and the final probe (EF) by applying the formula CF=max(EF, AC).

On the second opportunity, the same evaluation system will be used, with the same qualification of the continuous evaluation, but with the final grade corresponding to the second opportunity, which will be a final written exam of the same type as the first one.

As stated in the memorandum of the University of Santiago de Compostela for the studies of Master degree, that is, 24 hours of face-to-face work in the classroom (of which 18 hours are of theoretical lectures, 5 hours are practical work and 1 hour of tutorial work), 36 hours of personal work for the student (28 of which are of personal study, 5 are devoted to writing exercises and the 3 remainder hours are devoted to other types of practical work).

It is recommended to have a deep knowledge of Linear Algebra and the topology of Metric Spaces as well as a knowledge of Measure Theory and Ordinary Differential Equations.