Spring 2007
TMA4230: Functional Analysis

Remark: This is the course webpage! It is continously updated, so you should check it regularly. This includes the possible changes in exercises as well!

Lecturer:
Kamran Reihani

Main Book:
Erwin Kreyszig: Introductory functional analysis with applications.

H. Brezis. Analyse fonctionnelle.
Y. Eidelman, V. Milman, A. Tsolomitis. Functional analysis, An introduction.
J. Conway. A course in functional analysis.
M. Reed, B. Simon. Functional analysis.
W. Rudin. Functional analysis.
G. Pedersen. Analysis now.

H. Hanche-Olsen. Assorted notes on functional analysis. (2006-05-11 version) We will mainly focus on these notes for spectral theory.

Evaluation:
A midterm test, on March 19, will be written and counts 20%. You will be asked about what was taught before the winter break (14 lectures).
The final exam, on May 18, will be oral and counts 80%.

Important remark:
The main source for the exams will be what is taught in the classroom, with the exercises being important.
Basically, We intend to cover the Chapters 4, 7 and 9 of Erwin Kreyszig. Nevertheless, I am using some extra
matters form the other books and references given above, specially from H. Brezis, which is written in French.
Therefore, if you miss some lectures, the easiest way is to burrow your classmates' notes and make a copy.
Don't forget to solve the exercises!

Lecture 1 (08.01)
Introduction, Zorn's lemma.
Exercises: 2, 4, 8, 10 of Section 4.1.

Lecture 2 (11.01)
Hahn-Banach theorem.
Exercises: 3, 6, 8 of Section 4.2, and these.

Lecture 3 (15.01)
Generalized Hahan-Banach theorem.
Exercises: 1, 9, 11, 15 of Section 4.3.

Lecture 4 (18.01)
First geometric form of the Hahn-Banach theorem, Exercise session.
Exercises: These.

Lecture 5 (22.01)
First geometric form of the Hahn-Banach theorem (contd.), second geometric form of the Hahn-Banach theorem.

Lecture 6 (25.01)
Krein-Milman theorem.
Exercises: These.

Lecture 7 (29.01)
Exercises: 2, 5, 8, 9, 10 of Section 4.5.

Lecture 8 (01.02)
Exercises: 2, 4, 8, 10 of Section 4.6.

Lecture 9 (05.02)
Reflexive spaces (contd.).
Exercises: This.

Lecture 10 (08.02)
Banach-Steinhaus theorem (the principle of uniform boundedness), some applications of the Banach-Steinhaus theorem.
Exercises: 7, 8, 10, 14 of Section 4.7.

Lecture 11 (12.02)
An application of the Banach-Steinhaus theorem to Fourier series.

Lecture 12 (15.02)
The Baire's category theorem.
Exercises: 2, 3, 4, 6 of Section 4.7, and these.

Lecture 13 (19.02)
The open mapping theorem.
Exercises: 2, 5, 6, 8 of Section 4.12.

Lecture 14 (22.02)
The closed graph theorem.
Exercises: 5, 6, 8, 11, 12 of Section 4.13, and this.

There will be no lectures during the weeks 9 and 10 due to the winter break.
Please prepare for the midterm exam and solve as many exercises as you can including those which are posted above gradually and in Kreyszig.

Lecture 15 (12.03)
The closed graph theorem (contd.), strong and weak convergence in normed spaces.
Exercises: 9, 11, 13, 15 of Section 4.13, and 2, 4, 5 of Section 4.8, and this.

Lecture 16 (15.03)
Weak convergence (contd.), covergence of sequences of operators.
Exercises: 6, 7, 8, 9, 10 of Section 4.8, and 3, 5, 6 of Section 4.9, and this.

Written midterm exam (19.03). Results.

Lecture 17 (22.03)
Covergence of sequences of operators. (contd.), weak* convergence of functionals.
Exercises: 4, 7, 8, 9, 10 of Section 4.9.

Lecture 18 (26.03)
Spectral theory: Spectrum of operators in normed spaces (finite dimensional case and basic concepts).
Exercises: 2, 3, 4, 5, 6, 10, 13, 15 of Section 7.1.

Lecture 19 (29.03)
Spectrum of operators in normed spaces (contd.), spectral theory of bounded linear operators on Banach spaces, the notion of a Banach algebra.
Exercises: 2, 3, 4, 6, 7, 8, 9 of Section 7.2, 1, 2, 4, 6, 8, 10 of Section 7.3, and 2, 3, 6, 9 of Section 7.6.

There will be no lectures until April 12 due to the Easter holidays.

Lecture 20 (12.04)
Spectrum in Banach algebras: invertibility in algebras, the spectral mapping theorem, geometric series in Banach algebras.
Exercises: 4, 5, 8, 9 of Section 7.4, and 2, 3, 7, 10 of Section 7.7.

Lecture 21 (16.04)
Spectrum in Banach algebras: geometric series in Banach algebras (contd.), invertibility in Banach algebras, compactness of the spectrum in Banach algebras,
holomorphy of the resolvent, non-emptiness of spectrum in Banach algebras (Gelfand's theorem).
Exercises: 1, 2, 3 of Section 7.5, 1, 2, 3 of Section 7.4, and this.

Lecture 22 (19.04)
Spectrum in Banach algebras: holomorphic functional calculus, spectral radius.
Exercises: 7 os Section 7.4, 4, 5, 6, 7, 8, 9, 10 of Section 7.5, and this.

Lecture 23 (23.04)
Spectral theory of selfadjoint operators: invertibility of normal operators, the polarization identity for sesquilinear forms.
Exercises: 2, 3, 4 of Section 9.1, and this.

Lecture 24 (26.04)
Spectral theory of selfadjoint operators: Cauchy-Schwarz inequality for non-negative Hemitian forms, spectrum of bounded selfadjoint operators.
Exercises: 6, 7, 8, 9, 10 of Section 9.1, 3, 8, 9, 10 of Section 9.2, and these.

Lecture 25 (30.04)
Spectral theory of selfadjoint operators: functional calculus.
Exercises: 1, 3, 4, 6, 7, 9, 10, 12 of Section 9.3, and this.

Lecture 26 (03.05)
Spectral theory of selfadjoint operators: functional calculus (contd.), projections.
Exercises: 5, 6, 8, 9, 10 of Section 9.5.

Lecture 27 (07.05)
Spectral theory of selfadjoint operators: the spectral theorem.
Exercises: 1, 2, 3, 4, 5, 7, 8, 9 of Section 9.9.

Oral final exam (18.05). See the details and results.