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Welcome to the course! The schedule for the course can be found in TimeEdit.
Extra lecture on Monday 11/12, 10.00-11.45, in MVL15
Extra lecture on Wednesday 6/12, 13.15-15.00, in MVL15
Questions to the oral exam
Assignment II: only problem 3 and 8 from Exercise II; problem 11 from Exercise II is moved to Assignment III.
Teachers
Examiner and lecturer: Lyudmila Turowska, Department of Mathematical Sciences, room L3025, tel: 7725341, e-mail: turowska ("vid") chalmers.se
Course literature
You can consult the following book: Gerald B. Folland, Real Analysis, Modern techniques and their Applications, Second Edition, John Wiley & Sons, 1999, Chapters 5-7 and parts of Chapter 4 . I will also produce Lecture Notes
Description of course
Functional analysis arose in early 20th century when the need became apparent to study whole classes of functions rather than individual ones. For example, the investigation of differential and integral equations arising in Physics concerns the study of maps from a set of functions into itself. The basic idea of functional analysis is to apply geometric methods to functions and function spaces. A function is considered as a point in a space, and this space will be a vector space usually of infinite dimension. Geometric objects like balls, and also convergence, are introduced in these spaces. Some important results are the Hahn-Banach theorem and Baire's theorem with consequences. Duality, weak convergence and Alaoglu's theorem are presented. Then an important part of the course is devoted to Hilbert spaces and orthogonality. The dual space of continuous functions and the Riesz representation theorem are discussed. Also included is a basic spectral theory of bounded linear operators. The course will consist of lectures and exercise sessions. The exercises will be taken from Folland's book and from sheets handed out during the course.
Preliminary Program
In the following (F) refers to Folland's book and (N) to Lecture Notes.
Lectures
| Week |
Chapter | Contents |
|---|---|---|
| 1 |
1-8(N), 5.1, 5.5(F) |
Vector spaces, Normed spaces, Inner product spaces |
| |
|
Completeness, Banach and Hilbert spaces |
| 2 |
6.1(F) 9(N) |
Lp-spaces |
| |
10-11(N), 5.1(F) |
Linear operators and linear functionals. Dual spaces |
| 3 |
11(N), 6.2(F) |
The dual of Lp-spaces |
| |
12(N), 5.2(F) |
The Hahn-Banach Theorem and its consequences |
| 4 |
5.3(F), 13(N) |
The Baire Category Theorem and its consequences |
| |
|
The Uniform Boundedness Principle |
| 5 |
5.4(F), 14(N) |
Weak and weak-* convergence |
| |
|
Alaoglu's Theorem |
| 6 |
5.5(F), 15(N) |
Hilbert spaces: Riesz-Frechet Theorem. Orthogonality |
| 7 |
7.3(F), 16(N) |
Dual of the space of continuous functions. The Riesz Representation Theorem |
| 8 |
17-18(N) |
Linear bounded operators: spectrum |
Recommended exercises
| Week |
Exercises |
|---|---|
| 1 |
Exercises I |
| 2 |
Exercises II |
| 3 |
Exercises III |
| 4 |
Exercises IV |
| 5 |
Exercises V |
| 6 |
Exercises VI |
| 7 |
Exercises VII |
Course requirements
The learning goals of the course can be found in the course plan.
Assignments
Deadlines for the hand-in exercises are on Tuesdays, the dates in the table below.
The hand-in exercises will be presented partly at Fridays lectures, three days after the date in the table.
| Day |
Exercises |
|---|---|
| 7/11 |
2,6,8(Exercises I) |
| 14/11 |
3, 8(Exercises II) |
| 21/11 |
11, 13 (Exercise II) 3,4 (Exercises III) |
| 28/11 |
6, (Exercises III), 1,4,8 (Exercises IV) |
| 5/12 |
1, 2, 5 (Exercises V) |
| 12/12 |
1, 2, 5 (Exercises VI) |
Examination
Examination consists of the hand-in exercises specified above combined with an oral exam at the end of the course
Examination procedures
In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.
Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal, where you also can read about what rules apply to examination at University of Gothenburg.
At the exam, you should be able to show valid identification.
After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.
At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of
the review will be announced here on the course homepage. Anyone
who can not participate in the review may thereafter retrieve and
review their exam at the
Mathematical Sciences Student office.
Check that you have the
right grades and score. Any complaints about the marking must be
submitted in writing at the office, where there is a form to fill
out.
At re-examination:
Exams are reviewed and retrieved at the
Mathematical Sciences Student office.
Check that you have the
right grades and score. Any complaints about the marking must be
submitted in writing at the office, where there is a form to fill
out.
Old exams