Functional Analysis

Following a brief review and slight extension of the prerequisited material as covered in introductory courses, the course starts with locally convex spaces. These are the most convenient topological vector spaces to work with (because the Hahn-Banach theorems still hold) and they are highly relevant in analysis, even in the study of normed spaces where one has a "better" topology to start with.

The next topic to be taken up is the Riesz-Schauder theory of compact operators on a Banach space. This theory has a wide range of applications, including the completeness statement about eigenfunctions in selfadjoint Sturm-Liouville problems.

The bounded operators on a Banach space are a concrete example of a so-called Banach algebra. This functional analytic structure occurs at quite a few places and is well worth the individual attention it will be given as the next topic in this course, together with the subclass of C*-algebras. As we will see, normal operators on a Hilbert space are best understood from a C*-algebra perspective and C*-algebras also provide a natural framework for Fredholm theory on Hilbert spaces, a topic which we will also cover.

"A Course in Functional Analysis" by John B. Conway. Second edition (1990), corrected fourth printing (1997). Graduate Texts in Mathematics, Vol. 96. Springer Verlag. ISBN: 978-0-387-97245-9. Price: 63 euro approx.

Tip 1: this title is available in Springer's Yellow Sales at a reduced price of 35 euro until July 31, 2008.

Tip 2: this title is also available as an International Edition at www.abebooks.com for very low prices indeed. These International Editions originate typically from China and India and they are softcover versions on a lower quality paper. However, textually such editions are identical with the regular editions and the International Edition of Conway's book has already profitably been used by participants of this course. If you should decide to use this edition, don't forget to order timely.

Basic knowledge of Banach and Hilbert spaces and bounded linear operators as is provided by introductory courses, and hence also of general topology and metric spaces. Keywords to test yourself: Cauchy sequence, equivalence of norms, operator norm, dual space, Hahn-Banach theorems, Baire category theorem, closed graph theorem, open mapping theorem, uniform boundedness principle, inner product and Cauchy-Schwarz inequality, orthogonal decomposition of a Hilbert space related to a closed subspace, orthonormal basis and Fourier coefficients, adjoint operator, orthogonal projection, selfadjoint/unitary/normal operators. You should be familiar with these notions and results at a workable level before you take this course, which is not suitable as a first acquaintance with functional analysis. Knowledge of compact operators or reflexivity (topics covered in some introductory courses) is not a prerequisite.

Measure and integration theory is not a formal prerequisite, an intuitive knowledge will (have to) do in the beginning of the course. However, if you are taking this advanced course in functional analysis and have not taken a course in measure and integration theory yet, then you are not in balance as an analyst and you should take such a course parallel to this one. Later on in this functional analysis we will assume that all participants are familiar with measure and integration theory at a workable level.

Homepage of the course: http://www.math.leidenuniv.nl/~mdejeu/FA_mastermath_2008-2009

PLEASE NOTE: in week 51 the course will be given in another room, F123.