Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OT | 4 | 0 |
All the previous courses of Calculus and Mathematical Analysis.
Good knowledge of Linear Algebra and Basic Topology is also important.
Explain the concepts and fundamental results of the Lebesgue integral in Euclidean space.
Present the methods of functional analysis, in the context of Banach spaces and Hilbert spaces.
The course consists of 3 blocks:
Theory of Measure, Banach Spaces and Hilbert Spaces.
1. Limitations of the Riemann integral.
2. Lebesgue measure. Abstract measure theory.
3. Lebesgue integral. Abstract integral theory. Limit vs. integral.
4. Fundamental Theorem of Calculus. Variable change theorem. Fubini-Tonelli theorem.
5. Integrals dependent on a parameter. Differentiating under the integral sign.
6. Normed spaces. Banach spaces. Characteristics.
7. Spaces of sequences. Spaces of functions. Spaces of measures.
8. Bounded linear operators. Norm of an operator. Topology of bounded linear operators.
9. Applications: Volterra's integral equation.
10. Open Mapping Theorem and Closed Graph Theorem. Uniform boundedness principle.
11. Dual topological of a normed space. Hahn-Banach theorem.
12. Hilbert spaces. Theorem of the Projection. Orthogonality
13. Hilbertian basis. Bessel inequality. Parseval's identity.
14. Fourier series. Riemann-Lebesgue lemma.
15. Compact operators. Sturm-Liouville problem.
This subject has 2 hours of theory and 1 of problems per week.
It also consists of a total of 6 hours of seminars throughout the course.
Although it is not compulsory, it is highly recommended to attend classes to ask questions and venture answers, even if they are incorrect.
Theory: we will develop the main results and put them in the context of future applications.
Problems: students will receive some lists of exercises that we will solve in problem classes.
Seminars: will serve to complement the contents of theory and problems.
Students will also have a few hours of consultation in the teacher's office, to consult questions, discuss methods, etc.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Exercices lessons | 14 | 0.56 | 2, 14, 15, 4, 13, 7, 5, 3, 12, 10, 6 |
Theorical lessons | 30 | 1.2 | 2, 14, 15, 4, 13, 7, 5, 3, 12, 10, 6 |
Type: Supervised | |||
Seminars | 6 | 0.24 | 2, 14, 15, 4, 13, 7, 5, 3, 12, 10, 6 |
Type: Autonomous | |||
Personal study | 92 | 3.68 |
During the course we will do an evaluation activity (two hours) for each block. It will consist in presenting the demonstration of some result, of a list established before the evaluation, and in the resolution of exercises.
Block 1. Measure Theory (30%)
Block 2. Banach Spaces (30%)
Block 3. Hilbert Spaces (30%)
The delivery of solved exercises, as the teacher did, indicating, complements (10%) the course evaluation.
On the day designated by the Coordination of the Degree as a Final Exam (or recovery), students who have not passed the course will take a make-up exam with all the course material. The maximum score that can be obtained in this recovery exam is 7.
ALL THE CONTENTS OF THE COURSE ARE EVALUABLE (THEORY, PROBLEMS, SEMINARS).
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Block 1. Measure Theory | 30% | 2 | 0.08 | 2, 14, 15, 1, 4, 13, 7, 8, 5, 3, 12, 11, 10, 6 |
Block 2. Banach Spaces. | 30% | 2 | 0.08 | 2, 14, 15, 1, 4, 13, 7, 8, 5, 3, 12, 11, 9, 10, 6 |
Block 3. Hilbert Spaces | 30% | 2 | 0.08 | 2, 14, 15, 1, 4, 13, 7, 8, 5, 3, 12, 11, 9, 10, 6 |
Delivery of exercises | 10% | 2 | 0.08 | 2, 14, 15, 1, 4, 13, 7, 8, 5, 3, 12, 11, 9, 10, 6 |
J. Bruna, Anàlisi Real, UAB Servei de Publicacions, 1996.
J.M. Burgués, Integració i càlcul vectorial, UAB Servei de Publicacions, segona edició, 2002.
S. Lang, Real and functional analysis, Graduate texts in mathematics, Springer, 1993.
W. Rudin Real and functional analysis, Alambra,1979.