TMA401 / MMA400 Functional analysis Autumn 26
Course PM
This page contains the program of the course: lectures and exercise sessions. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.
Program
The schedule of the course is in TimeEdit.
Lectures (Mondays and Fridays)
(Observe that on Mondays there are two sessions, morning 8-10 and afternoon 15-17. This is due to the tight schedules of both the students and the teacher. The Monday morning sessions will be most on theory whereas the afternoon sessions will have some examples and problem solving)
| Study Week | Chapter | Contents |
| 1 | DM1, K | Introduction, vector spaces, completeness |
| 2 | DM1 | Banach spaces, linear mappings, fixed point theory. |
| 3 | K2 | Fixed point theory (cont.), Lp-spaces |
| 4 |
K3.1, 3.4-3.5. DM3.1-3.4, 3.6 |
[K]. Lp-spaces (cont.), [DM] Hilbert spaces |
| 5 | DM 3.7, DM4.1-4.7 | Riesz Theorem. Linear operators on Hilbert spaces |
| 6 |
DM4.8-4.10, K4 |
Compact operators, spectral theory |
| 7 | DM5.8-5.10, K5 | Spectral Theory. Applications to ODE |
| 8 | Question session |
NOT INCLUDED:
[DM]: Chapt 2, Chapt 3.5], Chapt 4.11, Chapt 5.4-5.7, Chapt 5.11-5.12.
K: Part 2.4 (Starting from Theorem 2.4.3) -2.5. (Theorems 2.4.1 and Theorem 2.4.2 are included; Thm 2.4.1 is a general elementary statement and we shall
not prove it, Thm 2.4.2 is a criterion for compactness in C(K) and shall not be proved, but we shall illustrate it by examples.)
Lecture Notes.
Week 1-2 (a) (Normed spaces. Höder-Minskowski-ineq.)
Week 1-2(b) (Banach-spaces. Riesz-thm and non-compactness)
Week 2-3(a) (Lin. mapping, Fixed-pt-thm. Banach-Steinhauss-thm)
Week3-4 (Compactness in Banach spaces)
Week 4 (Hilbert Space -p. 1, Hilbert Space p. 2-9)
Week 5 (Hilbert space, bilinear forms)
Week 6 (Operators. Compact Operators-Chapt4)
Week 6-7(Compact-Op. Self-adj-Compact Op. Fri W6, MonMorning.W7)
Week 7-8 (Fredholm-ODE-Green Functions. Week7Monday/Friday+W8Monday.)
Week 8 (Green+Fixed-Pt-ODE)
Exercises (Thursdays) and Recommended exercises
Below is a list of recommended exercises. The exercise classes will be demonstration/discussion of the exercises with *.
| Week | Exercises |
| 1 | DM1: 1, 5, 9*, 13, 28, 30*, 36, 37, 40, 41*, 45 |
| 2 | K7.2: 2, 7,11, 12, 13, 17, 18*, 31*, 38 K7.3: 11, 16, 17, 21*,45, 66, 72, 78*, 89 |
| 3 | K7.4: 2, 6, 7* (Further question for discussion: Construct non-linear examples), 10, 14*, 16, 17*, 25 |
| 4 | [DM]Chapt2: 2.41*. K7.5: 9, 10, 11, 12, 15, 19*, 22, 31*, 37* |
| 5-6 | K7.6: 2, 3*, 4, 5, 6, 9, 10, 12 |
| 6 | [DM] Chapt 3: 51*. K7.6: 3*, 9*, 14, 16, 17, 25, 26, 29, 33, 39. Mini-exam exercises. |
| 7-8 | Week 7: K7.6:84*. K7.7: 1(a)* 2, 12, 16, 20*, 25; Week 8, K7:1(b)(c), 6. Earlier Exam problems |
(Dugga/Mini-Exam Study Week 5.)
*DMx refers to Chapter x of L. Debnath/P. Mikusinski: Hilbert Spaces with Applications.
**Kx refers to the Chapter x of the lecture notes by P. Kumlin (see the course PM.)
Examination
The exam will consist of solving concrete problems and proving the existing theorems (see below) and proving new theorems using the proved theorems in the course literature. Here is a list of the main theorems whose statements/proofs/applications may appear in the exam.
Computer labs
There are no computer labs in this course.
Reference literature:
- Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
- Physical Modeling in MATLAB 3/E, Allen B. Downey
The book is free to download from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modeling and simulation of physical systems.
Course summary:
| Date | Details | Due |
|---|---|---|