The final exam will be on June 13, Tuesday, in room 3733 at 10:00.

Every student will get a ticket with two questions. The first one will be of theoretical character and the second one will be a problem chosen from the home exercises.

The main goal is to give an introduction to the basics of functional analysis and operator theory, and to some of their (very numerous) applications.

The lectures will be on Tuesdays between 10:15-12:00, room 3733, institutionen för matematik. First lecture is on January 17.

• Banach spaces. Bounded operators.
• The Banach-Steinhaus theorem.
• The open mapping theorem.
• The closed graph theorem.
• Dual spaces. The Hahn-Banach theorem.
• Dual spaces to L^p and C(X).
• Weak topologies. The Banach-Alaoglu theorem.
• Adjoint operators.
• Compact operators and their spectrum.
• Fredholm alternative.
• Hilbert spaces and operators on Hilbert spaces.
• The spectral theorem for self-adjoint operators on Hilbert spaces.
• Fredholm determinant.
• Unbounded operators.

Lecture 1 and Exercises: pdf , text ; Homework results: pdf .

Lecture 2 and Exercises: pdf , text ; Homework results: pdf .

Lecture 3 and Exercises: pdf , text ; Homework results: pdf ; Solution of Problem 4: pdf .

Lecture 4 and Exercises: pdf , text ; Homework results: pdf .

Lecture 5 and Exercises: pdf , text ; Homework results: pdf .

Lecture 6 and Exercises: pdf , text ; Homework results: pdf .

Lecture 7 and Exercises: pdf , text ; Homework results: pdf .

Lecture 8 and Exercises: pdf , text ; Homework results: pdf .

Lecture 9 and Exercises: pdf , text ; Homework results: pdf .

Lecture 10 and Exercises: pdf , text ; Homework results: pdf .

Lecture 11 and Exercises: pdf , text ; Homework results: pdf .

Lecture 12 and Exercises: pdf , text ; Homework results: pdf .

Lecture 13 and Exercises: pdf , text ; Homework results: pdf .

Lecture 14 and Exercises: pdf , text ; Homework results: pdf .

Lecture 15 and Exercises: pdf , text ;

Avner Friedman, Foundations of Modern Analysis, Dover Publications, Inc., 1982.

Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional Analysis, Academic Press, 1980 (First part of a multi-volume textbook on mathematical physics. It gives a firm functional analysis background for use in the mathematical physics).

M.S.Birman and M.Z.Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D.Reidel Publishing Company, 1987.

Peter Lax, Functional Analysis, John Wiley & Sons, 2002 (Includes sections on the spectral resolution and spectral representation of self adjoint operators, invariant subspaces, strongly continuous one-parameter semigroups, the index of operators, the trace formula of Lidskii, the Fredholm determinant, and more).