Real and Functional Analysis
Code: 100110Credits: 6
| Degree programme | Type | Course |
|---|---|---|
| Mathematics | OP | 4 |
Contact lecturer
- Name :
- Joan Orobitg Huguet
- Email :
- joan.orobitg@uab.cat
Group languages
You can consult this information at the end of the document.
Prerequisites
All the previous courses of Calculus and Mathematical Analysis.
Good knowledge of Linear Algebra and Basic Topology is also important.
Objectives
Explain the fundamental concepts and results of measure theory (example, Lebesgue integral).
Present the methods of functional analysis, in the context of Banach and Hilbert spaces.
Learning outcomes
- Effectively use bibliographies and electronic resources to obtain information.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
- Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Generate innovative and competitive proposals for research and professional activities.
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Confidently deal with the most important Hilbert spaces and know how to apply the basic theory of Functional Analysis to them.
- Understand the concept of R^n measurement and its construction process.
- Understand the language and in-depth demonstrations of some advanced mathematical analysis theorems.
- Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
- Understand the nature of the Lebesgue integral and its advantages over the Riemann integral.
- Formulate conjectures and devise strategies to confirm or reject said conjectures
- Devise demonstrations of mathematical results in the field of mathematical analysis.
Contents
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.
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Theorical lessons | 30 | 1.2 | 1, 2, 4, 5, 9, 10, 11, 12, 13, 14, 15 |
| Personal study | 92 | 3.68 | 2, 3, 5, 6 |
| Exercices lessons | 14 | 0.56 | 1, 2, 4, 5, 9, 10, 11, 12, 13, 14, 15 |
| Seminars | 6 | 0.24 | 1, 2, 4, 5, 9, 10, 11, 12, 13, 14, 15 |
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.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| Delivery of exercises | 10% | 2 | 0.08 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 |
| Block 2. Banach Spaces. | 30% | 2 | 0.08 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 |
| Block 1. Measure Theory | 30% | 2 | 0.08 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 |
| Block 3. Hilbert Spaces | 30% | 2 | 0.08 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 |
During the course we will do an assessment activity (two hours) for each block. It will consist of solving exercises and/or presenting the proof of some result, from a list established before the assessment.
Block 1. Measure Theory (30%) All students who take this exam may no longer be classified as NOT EVALUABLE. Any student who has not taken this exam will be recorded as NOT EVALUABLE for academic purposes and will not have the right to retake it (except for duly justified reasons, in which case a retake exam will be allowed).
Block 2. Banach Spaces (30%) Any student who has not taken this exam will not have the right to retake it (except for duly justified reasons, in which case a retake exam will be allowed).
Block 3. Hilbert Spaces (30%) Any student who has not taken this exam will not have the right to retake it (except for a duly justified reason, in which case a retake exam will be allowed).
The submission of solved exercises, as indicated by the professor, complements (10%) the course evaluation. On the day designated by the Degree Coordination as the Final Exam (or retake), students who have not passed the subject will take a retake exam covering all the course material. The maximum score that can be obtained in this retake exam is 7. ALL THE SUBJECT CONTENTS ARE ASSESSABLE (THEORY, PROBLEMS, SEMINARS).
For each evaluation activity, a place, date and time of review will be indicated in which students will be able to review the activity with the professor. In this context, complaints may be made about the grade of the activity, which will be evaluated by the professor responsible for the subject. Students who do not attend this review will not be able to review this activity later.
SINGLE EVALUATION: Students who have opted for the single evaluation method must take a final test that will consist of a theory exam where they must develop a topic and/or answer a series of short questions. They will then have to take a problem/practice test where they must solve a series of exercises similar to those worked on in the Classroom Practice/Problems 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/practice exam for 70%. If the final grade does not reach 5, the failed student has another opportunity to pass the subject through the retake exam that will be held on the date set by the degree coordination. In order to be able to take the retake, a minimum grade of 3.5 must be obtained. The maximum score that can be obtained in this retake exam is 7. The review of the final grade follows the same procedure as for the continuous assessment. The proposed teaching methodology and assessment may undergo some modification depending on the restrictions on face-to-face learning imposed by the health authorities.
"The commission of any irregularity in an assessment act (academic fraud, plagiarism or improper use of AI, unless such use is expressly authorized in the teaching guide), which may lead to a significant variation in the grade, means that this act will be graded with a 0. In the event that the teaching guide provides that in order to pass the subject it is an essential requirement to have obtained a minimum grade in this assessment act or that several irregularities occur in the assessment acts of the same subject, the final grade for this subject is 0. Apart from this, a disciplinary process may be initiated against the student who incurs any of these irregularities."
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.
Bibliography
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.
Walter Rudin, Functional analysis, Alambra,1979.
Walter Rudin, Real and Complex Analysis, 1987, tercera edició.
Software
We will not use any specialized software.
Course groups and languages
The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE
| Type of teaching | Group | Language | Semester | Shift |
|---|---|---|---|---|
| (TE) Theory | 1 | Catalan | first semester | morning-mixed |
| (PAUL) Classroom practices | 1 | Catalan | first semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan | first semester | afternoon |