Spring 2007
MAT4350: Functional Analysis

Main book:
Y. Eidelman, V. Milman, A. Tsolomitis. Functional analysis. An introduction.

Additional literature:
G.K. Pedersen. Analysis now.
J.B. Conway. A course in functional analysis.
W. Rudin. Functional analysis.


Lecture 1 (15.01)
The Baire category theorem. The Banach open mapping theorem. The Banach-Steinhaus theorem.
Sections: 9.1, 9.2, 9.4.
Exercises: these four.

Lecture 2 (22.01)
Schauder basis. The Hahn-Banach theorem.
Sections: 9.5, 9.6.
Exercises: 22.

Lecture 3 (29.01)
The Hahn-Banach theorem. Separation of convex sets.
Sections: 9.6, 9.7.
Exercises: these two.

Lecture 4 (05.02)
Weak topologies.
Sections: 9.7, 9.8.
Exercises: 29 and these five.

Lecture 5 (12.02)
The Krein-Milman theorem.
Sections: 9.9.
Exercises: these five.

Lecture 6 (19.02)
Uniformly convex spaces. Analytic vector-valued functions.
Sections: 10.3.

Lecture 7 (26.02)
Banach algebras. Spectrum. Multiplicative functionals.
Sections: 10.1, 10.3.
Exercises: these four.

Lecture 8 (05.03)
Spectrum. The Gelfand transform.
Sections: 10.2-10.5.
Exercises: these three.

Lecture 9 (12.03)
Wiener's Tauberian theory.
Exercises: these two.

Lecture 10 (19.03)
C*-algebras.
Sections: 10.6.
Exercises: 15 and these three.

Lecture 11 (26.03)
Spectral theorem.
Sections: 10.7.
Exercises: these three.

Lecture 12 (16.04)
Unbounded operators. The closed graph theorem. Self-adjoint operators.
Sections: 9.3, 11.4, 11.5.

Lecture 13 (23.04)
Spectral theorem for unbounded self-adjoint operators. Symmetric operators.
Sections: 11.1, 11.3, 11.5, 11.6.
Exercises: these five.

Lecture 14 (30.04)
The Friedrichs extension. Self-adjoint operators and one-parameter unitary groups.
Exercises: these two.

Lecture 15 (23.05)
Commuting self-adjoint operators. Nelson's example.