ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Statistics, Mathematical Analysis and Optimisation
Areas: Mathematical Analysis
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable
The aim is to present the elementary principles of functional analysis with special emphasis on Hilbert spaces. Thus, the fundamental properties of Hilbert spaces, their geometry and linear & continuous mappings (operators) in Hilbert spaces are studied. The concepts and basic results of spectral theory for operators in Hilbert spaces are also introduced and some of its multiple applications are commented on.
1. Examples of Hilbert spaces. Orthogonal projections. Angles. (1 hour)
2. Hilbert spaces. (2 hours)
3. Normed spaces. Topological vector spaces. (2 hours)
4. Examples. (1 hour)
5. Orthogonal projection theorem. Consequences. (2 hours)
6. Riesz representation theorem. (1 hour)
7. Isomorphism between Hilbert spaces. (1 hour)
8. Adjoint operator. (1 hour)
9. Separable Hilbert spaces. (1 hour)
10. Characterization of the finite / infinite Hilbert spaces. (1 hour)
11. Theorem of Ascoli-Arzelà. (1 hour)
12. Operators. Adjoint operators. (2 hours)
13. Spectrum of an operator. (2 hours)
14. Compact operators. (1 hour)
15. Spectral theorem. (2 hours)
16. Theorem of Lax-Milgram. Applications to differential equations. (2 hours)
17. Integral equations. (2 hours)
18. Quantum mechanics. (1 hour)
19. Differentiability in Hilbert spaces. (1 hour)
20. Variational calculus. (1 hour)
Basic Bibliography:
L. Abellanas, A. Galindo, Espacios de Hilbert (Geometría, Operadores, Espectros), EUDEMA, 1987.
H. Brezis, Análisis Funcional, Alianza Universidad Textos, 1984.
Complementary bibliography:
B. Cascales, J.M. Mira, J. Origuela, y M. Raja, Análisis Funcional. Electrolibris : Real Sociedad Matemática Española, 2013.
J. Cerdá, Linear Functional Analsyis, American Mathematical Society, 2010.
H. Brezis, Functional Analysis Sobolev Spaces and Partial Differential Equations. Springer, 2011.
GC1 - Know the most important concepts, methods and results of different areas of Mathematics, together with a certain historical perspective on their development.
GC2 - Collect and interpret the data, information and relevant results; obtain conclusions and issue reasoned reports on scientific, technological or other problems that require the use of mathematical tools.
GC4 - Communicate in writing and orally about the knowledge, procedures, results and ideas in Mathematics to specialized as well as non-specialized public.
GC5 - Organize time and resources to study and learn new knowledge and techniques in any scientific or technological discipline.
SC1 - Understand and use mathematical language.
SC2 - Knowledge of rigorous proofs of some classical theorems from different areas of Mathematics.
SC3 - Derive proofs for mathematical results, formulate conjectures and develop strategies to verify the same.
SC4 - Identify errors in propositions and propose proofs or counterexamples.
SC5 - Assimilate the definition of a new mathematical object/concept, relate it to the other already known objects/concepts, and apply it in different contexts.
SC6 - Knowing how to abstract the properties and substantial facts of a problem, distinguishing them from those that are purely occasional or circumstantial.
TC1 - Use bibliography and search tools for general and specific bibliographic resources of Mathematics, including Internet access.
TC2 - Optimally manage work time and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision making.
TC3 - Check or reasonably reject the arguments of other people.
TC5 - Read scientific texts in the original languages as well as other languages of importance in scientific field, such as English.
(GC: general competence; SC: specific competence; TC: trans-disciplinary competence)
The general methodological indications established in the Guidelines for Bachelor of Mathematics (USC) will be followed.
The teaching is programmed to be conducted in master classes, interactive classes and tutorials in small groups. In master classes, the essential contents of the subject will be presented; in interactive classes, problems and exercises previously proposed by the teacher will be solved; and tutorials in small groups will be dedicated to discussion and debate with the students. It will help in elevating the participation and critical thinking of students, especially in the interactive classes.
The final exam will consist of the resolution of theoretical and practice questions similar to those studied during the semester.
The competences associated to the declarative contents of the subject (GC1, SC1, SC2, SC3, SC4, SC5, SC6) will be assessed in the final exam.
Continuous assessment: it will consist of two examinations to be taking during lecture time. The exact date of the examination will be announced in advance.
Calculation of the final grade: The numerical grade of the opportunity will be computed as max{E,0.4C+0.6E} where E is the grade of the final exam of the opportunity (which will take place at the dates indicated by the Faculty) and C is the average of the continuous assessment.
Those students who do not participate at the final exam of a given opportunity will be scored as “not presented” in that opportunity.
In those cases of fraudulent behavior regarding assessments the precepts gathered in the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” will be applied.
ON-SITE WORK AT CLASSROOM
Blackboard classes in large group (28 hours)
Seminaries (14 hours)
Laboratories (14 hours)
Tutorials (2 hours)
Evaluation activities (5 hours)
TOTAL: 63 hours
PERSONAL WORK OF THE STUDENT: 87
Students are advised to have passed certain subjects viz. Vector Calculus & Lebesgue Integration, General Topology and Fourier Series & Introduction to partial differential equations.
Regular (daily) and rigorous work is expected. It is basic to take part actively in the learning process of the subject. To attend regularly to lectures both theoretical and practical, to participate in them, to formulate questions as well.
Juan José Nieto Roig
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813177
- juanjose.nieto.roig [at] usc.es
- Category
- Professor: University Professor
| Monday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 08 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLIS_01 | Spanish | Classroom 08 |
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 08 |
| Thursday | |||
| 10:00-11:00 | Grupo /CLIL_01 | Spanish | Computer room 4 |
| 11:00-12:00 | Grupo /CLIL_02 | Spanish | Computer room 4 |
| 12.21.2022 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 06.16.2023 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |