Functional Analysis 2008

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.

• 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.

Lectures and exercises - week 1
Lectures and exercises - week 2
Lectures and exercises - week 3
Lectures and exercises - week 4
Lectures and exercises - week 5
Lectures and exercises - week 6
Lectures and exercises - week 7
Lectures and exercises - week 8
Lectures and exercises - week 9
Lectures and exercises - week 10


Books

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

Other books

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).