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. Topological vector spaces. (2h)
2. Banach spaces (8h)
2.1. Definition and examples.
2.2. Linear operators in normed spaces.
2.3. Fundamental theorems in the Theory of Banach spaces.
2.4. Dual spaces.
2.5. Finite / infinite dimension characterizations.
2.6. Ascoli-Arzelà’s Theorem.
3. Hilbert spaces. (9h)
3.1. Definition and examples.
3.2. The orthogonal projection theorem.
3.3. Riesz’s representation theorem.
3.4. Isomorphisms in Hilbert spaces. Adjoint operators.
3.5. Separable Hilbert spaces.
4. Hilbert space operators. (6h)
4.1. Spectrum of an operator.
4.2. Spectral theorem.
4.3. Projections.
4.4. Examples of operators.
4.5. Applications.
Basic Bibliography:
Friedman, A. Foundations of Modern Analysis, 2ª ed. New York: Dover, 1982. (1202 213)
Gohberg, I. e Goldberg, S. Basic Operator Theory, 1ª ed. Boston: Birkhäuser, 1981. (1202 265 A)
Gohberg, I., Goldberg, S. e Kaashoek, M. A. Basic Classes of Linear Operators, 1ª ed. Birkhäuser, 2003. (47 241)
Kreyszig, E. Introductory Functional Analysis with applications, John Wiley & Sons, 1978. (1202 264 A)
Megginson, H. An introduction to Banach Space Theory, Springer 1998. (1202 361 A)
Complementary bibliography:
Bollobás, B. Linear analysis: an introductory course, 2ª ed. Cambridge: Cambridge Mathematical Textbooks, 1999. (1202 435)
Brezis, H. Análisis Funcional, Alianza Universidad Textos 1984. (1202 37 C)
Conway, J.B. A Course in Functional Analysis, Springer 1990. (1202 289 A)
Rudin, W. Real and Complex Analysis, 3ª ed. New York: McGraw-Hill, 1987. (1202 20 F)
Vera López, A. Un curso de Análisis Funcional. Teoría y problemas, AVL 1997. (1202 195 A)
On-line materials (accesible from Springer Link, see https://www.youtube.com/watch?v=t8hPlEwNFLg&feature=emb_logo )
Cohen, D. W. An Introduction to Hilbert Space and Quantum Logic.
URL https://link.springer.com/book/10.1007/978-1-4613-8841-8
Friedrichs, K. O. Spectral Theory of Operators in Hilbert Space.
URL https://link.springer.com/book/10.1007/978-1-4612-6396-8
Halmos, P., A Hilbert space problem book.
URL https://link.springer.com/book/10.1007/978-1-4684-9330-6
Gohberg, I. y Goldberg, S., Basic Operator Theory.
URL: https://link.springer.com/book/10.1007/978-1-4612-5985-5
Gohberg, I., Goldberg, S. y Kaashoek, M. A., Basic Classes of Linear Operators.
URL: https://link.springer.com/book/10.1007/978-3-0348-7980-4
Krall, A. M. Hilbert Space, Boundary Value Problems and Orthogonal Polynomials.
URL: https://link.springer.com/book/10.1007/978-3-0348-8155-5
Kubrusly, C. The Elements of Operator Theory.
URL https://link.springer.com/book/10.1007/978-0-8176-4998-2
Lal Vasudeva, H. Elements of Hilbert Spaces and Operator Theory.
URL: https://link.springer.com/book/10.1007/978-981-10-3020-8
Muscat, J. Functional Analysis.
URL: https://link.springer.com/book/10.1007/978-3-319-06728-5
Schmüdgen, K. Unbounded Self-adjoint Operators on Hilbert Space.
URL: https://link.springer.com/book/10.1007/978-94-007-4753-1
Siddiqi, A. Functional Analysis and Applications. URL:
https://link.springer.com/book/10.1007/978-981-10-3725-2
Steeb, W. Hilbert Spaces, Wavelets, Generalised Functions and Modern Quantum Mechanics.
URL: https://link.springer.com/book/10.1007/978-94-011-5332-4
Sunder, V. S. Operators on Hilbert Space.
URL: https://link.springer.com/book/10.1007/978-981-10-1816-9
Weidmann, J. Linear Operators in Hilbert Spaces.
URL https://link.springer.com/book/10.1007/978-1-4612-6027-1
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 three written examinations to be taking during lecture time. The exact date of the examination will be announced in advance.
Computation 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.
Both the continuous evaluation tests and the final exam will be the same in all the exhibition and interactive teaching groups of the subject.
WORK IN CLASSROOM
Lecture hours (26 hours)
Hours of interactive seminars (13 hours)
Interactive laboratory hours (13 hours)
Tutoring in small groups (2 hours)
Total classroom attendance hours: 54.
PERSONAL WORK: About 96 h depending on the person and her background.
Students are advised to have passed certain courses 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.
Fernando Adrian Fernandez Tojo
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- fernandoadrian.fernandez [at] usc.es
- Category
- Professor: University Lecturer
Victor Cora Calvo
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- victor.cora.calvo [at] usc.es
- Category
- Xunta Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 09 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom 09 |
| 11:00-12:00 | Grupo /CLIS_01 | Spanish | Classroom 09 |
| Thursday | |||
| 11:00-12:00 | Grupo /CLIL_01 | Spanish | Classroom 09 |
| 01.08.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 06.13.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |