Degree | Type | Year |
---|---|---|
2500149 Mathematics | OT | 4 |
You can view this information at the end of this document.
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.
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 |
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.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Block 1. Measure Theory | 25% | 2 | 0.08 | 2, 14, 15, 1, 4, 13, 7, 8, 5, 3, 12, 11, 10, 6 |
Block 2. Banach Spaces. | 25% | 2 | 0.08 | 2, 14, 15, 1, 4, 13, 7, 8, 5, 3, 12, 11, 9, 10, 6 |
Block 3. Hilbert Spaces | 25% | 2 | 0.08 | 2, 14, 15, 1, 4, 13, 7, 8, 5, 3, 12, 11, 9, 10, 6 |
Delivery of exercises | 25% | 2 | 0.08 | 2, 14, 15, 1, 4, 13, 7, 8, 5, 3, 12, 11, 9, 10, 6 |
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 (25%)
Block 2. Banach Spaces (25%)
Block 3. Hilbert Spaces (25%)
The delivery of solved exercises, as the teacher did, indicating, complements (25%) 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).
For each assessment activity, a place, date and time of review will be indicated in which
students can review the activity with the teaching staff. In this context
about the grade of the activity, which will be evaluated by the teaching staff responsible for
the subject. Students who do not attend this review will not be able to review the activity at a
later date.
One-time evaluation:
Students who have opted for the single assessment mode must take a final exam consisting of
a theory exam where they will have to develop a topic and/or answer a series of short
questions. They will then have to take a problem/practical test where they will have to solve a
series of exercises similar to those that have been worked on in the Practical/Classroom
Problem sessions.
The grade will be the weighted average of the two previous activities, where the theory exam
will account for 30% of the grade and the problem/practical exam for 70%.
If the final grade does not reach 5, the failed students have another opportunity to pass the
subject by means of the make-up exam that will be held on the date set by the degree
coordination. In order to take the make-up exam, students must have obtained a minimum
grade of 3.5.
The review of the final grade follows the same procedure as for continuous assessment.
The proposed teaching methodology and assessment may undergo some modification
depending on the restrictions imposed by the health authorities.
This English version of the guide is a translation of the Catalan version. In the event of any
discrepancy between the two, the correct version for all purposes is the Catalan version.
J. Bruna, Anàlisi Real, UAB Servei de Publicacions, 1996.
J.M. Burgués, Integració i càlcul vectorial, UAB Servei de Publicacions, segona edició, 2002.
J. L. Cerdà Martín, Análisis Real, Col·lecció UB 23, segona edició, 2000.
J. L. Cerdà Martín, Introducció a l'Anàlisi Funcional, Textos Docents 280, Publicacions i edicions UB, 2005.
W. Rudin, Functional analysis, Alambra,1979.
We will not use any specialized software.
Name | Group | Language | Semester | Turn |
---|---|---|---|---|
(PAUL) Classroom practices | 1 | Catalan | first semester | morning-mixed |
(SEM) Seminars | 1 | Catalan | first semester | afternoon |
(TE) Theory | 1 | Catalan | first semester | morning-mixed |