Spring 2007

MAT4350: Functional Analysis

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.