Latest news
Welcome to the course.
The schedule for the course can be found via the link to webTimeEdit top of the page.

Material from the last lecture is added to the Lecture notes. The part on dual of C_0(X) is corrected.
Questions to the oral exam

Extra lecture 23/11, at 13.15 in MVH11

Lecture from 8/11 is moved to 9/11, at 13.15 in MVL15!!!
Teachers
Examiner and lecturer: Lyudmila Turowska, Department of Mathematical Sciences, room L3025, tel: 7725341, e-mail: turowska ("vid") chalmers.se
Course litterature
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. 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 Programme
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)
Lp-spaces

9-11(N), 5.1(F)
Linear operators and linear functionals. Dual spaces.
3
11(N), 6.2(F)
The dual of L_p

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)
Dual of the space of continuous functions. The Riesz Representation Theorem
8

Linear 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



Hand-in exercises
Deadlines for the hand-in exercises are on Tuesdays, the dates in the table below.
The hand-in exercises will be presented at Fridays lectures, three days after the date in the table.

Day Exercises
9/11
5,7 (Exercise I)
15/11
3, 8, 11(Exercise II)
22/11
3, 4, 6 (Exercise III)
29/11
1, 4, 8(Exercise IV)
6/12
1, 2, 5(Exercise V)
13/12
1,2,5(Exercise VI)


Course requirements

Assignments


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.
At the link Scedule you can find when exams are given for courses at University of Gothenburg.
At the exam, you should be able to show valid identification.
Before the exam, it is important that you report that you want to take the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, so sign up via GU's Student Portal.

You can see your results in Ladok by logging on to the Student portal.

At the annual examination:
When it is practical a separate review is arranged. The date of the review will be announced here on the course website. Anyone who can not participate in the review may thereafter retrieve and review their exam on Mathematical sciences study expedition, Monday through Friday, from 9:00 to 13:00. 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 picked up at the Mathematical sciences study expedition, Monday through Friday, from 9:00 to 13:00. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.
Old exams