Functional Analysis 2008 (SF 2707)

Ari Laptev

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.

First lecture will be on January 18 between 15-17 in the seminar room 3733, institutionen för matematik.
We continue our lectures every second Monday starting from January 28 between 10-15 with one hour break for lunch in the seminar room 3721 Institutionen för matematik.

• 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 determinants.
• Unbounded operators.

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

M. Reed and B. 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.

P. 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).