Functional analysis II
Summary
We introduce locally convex vector spaces. As an example we treat the space of test functions and the space of distributions. In the second part of the course, we discuss differential calculus in Banach spaces and some elements from nonlinear functional analysis.
Content
- Locally convex vector spaces.
- Test functions and the dual space of distributions.
- Fréchet-derivative.
- Implicit function theorem and consequences on Banach spaces.
- Fixed-point theorems.
- Introduction to degree theory (if time permits).
Keywords
Locally convex vector spaces, test functions and distributions, analysis on Banach spaces, nonlinear functional analysis.
Learning Prerequisites
Required courses
Analysis I-IV, Linear Algebra I-II, Metric and topological spaces, Functional analysis I.
Important concepts to start the course
Basic notions from topology, Banach spaces, differential calculus in finite dimensions, Lebesgue integration.
Learning Outcomes
By the end of the course, the student must be able to:
- Formulate the definitions and results of the lectures
- Apply the concepts learned in class to concrete problems
- Analyze problems related to the topics treated in the course
- Choose an appropriate method to solve a given problem
- Prove some elementary statements about the topics of the course
- Solve exercises on the topics
Teaching methods
Weekly lectures (on blackboard) and exercise sessions with assistant.
Expected student activities
Attending the lectures and solving the exercises.
Assessment methods
Written exam.
Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.
Supervision
| Office hours | Yes |
| Assistants | Yes |
| Forum | Yes |
| Others | Office hours by appointment. |
Resources
Virtual desktop infrastructure (VDI)
No
Bibliography
W. Rudin, Functional Analysis. McGraw-Hill, INc., 2nd ed. 1991.
M. Reed and B. Simon. I: Functional analysis. Vol. 1, Orlando Academic Press, 1980.
K. Deimling, Nonlinear Functional Analysis, Springer 1985.
Ressources en bibliothèque
- Functional Analysis / Rudin
- Nonlinear Functional Analysis / Deimling
- Functional analysis. Vol. 1 / Reed
Notes/Handbook
Lecture notes will be available in moodle.
Moodle Link
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Functional analysis II
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional