# Functional Analysis

 Credits 8 credit points Instructors Verduyn Lunel, S.M. (Universiteit Leiden) E-mail verduyn@math.leidenuniv.nl Prerequisites Apart from (advanced) calculus (including elementary differential equations and a little bit of knowledge of functions of a complex variable) no prior knowledge is assumed (only simple topological and algebraic concepts are used and these are introduced and proved as needed; no measure theory is employed or mentioned). But the speed with which the material will be covered relies on the assumption that the students have already some  familiarity with (open, closed and compact sets in) metric spaces, with Cauchy sequences and completeness, with equivalence classes, with Fourier series and with eigenvalues and eigenvectors of matrices (and the linear mappings these represent). Students that have taken an introductory functional analysis course (as offered by several universities as part of the bachelor) will certainly profit from it.  Some subjects (e.g., spectral theory of compact self-adjoint operators) will return in this course at a higher level of abstraction. It is not a formal prerequisite that such an introductory course has been taken. Indeed, the Schechter book is self contained and students that didn't follow an introductory course should simply invest a bit more (mental) energy and time to "digest" the material. Aim To become acquainted with the basic elements of analysis in infinite dimensional spaces, while focussing on normed vector spaces (in particular Banach - and Hilbert spaces) and linear operators acting between them. Description Historically, functional analysis grew out of the need to understand and formalize the "linear algebra of infinitely many unknowns" related to integral equations and boundary value problems for linear differential equations. It developed into a beautiful subject which can be motivated and studied for its own sake. In this course we restrict attention to normed vector spaces. If such a space is complete, we call it a Banach space. If the vector space admits a scalar (or, inner)  product and the norm of any element equals the square root of  the scalar product of the element with itself, we call it a Hilbert space.  An assignment of a number to each element of a normed vector space is called a functional. Of particular interest are functionals which are bounded and  linear. One can provide the vector space of such functionals with a norm that turns it into a Banach space, called the dual (or, conjugate) space. We shall learn how to characterize and represent elements of the dual space when, for instance, the original space is a Hilbert space. We also present (versions of) the Hahn-Banach theorem about extending a linear functional which is originally only defined on a subspace. Next we turn attention to linear operators mapping one normed vector space into another (or the same) normed vector space. We introduce the adjoint operator and the inverse operator and present the Closed Graph Theorem, the Uniform Boundedness Principle (Banach-Steinhaus) as well as the Open Mapping Theorem. Spectral analysis is, perhaps, THE main tool for analyzing linear operator equations (it extends the characterization of a matrix in terms of its eigenvalues and eigenvectors to the infinite dimensional situation). Its treatment is preceded by chapters on the Riesz theory for compact operators and on Fredholm operators and followed by a chapter on unbounded operators. The remaining chapters are called : Banach algebras, semigroups (of operators), Hilbert space (and normal operators), bilinear forms, selfadjoint operators, measures of operators (like measure of noncompactness), examples and applications. At this point it is not yet decided what selection from this material will be covered in the course (and the students may influence the decision!). Organization Each class there will be three 45 minutes time slots. The middle one is devoted to making exercises (except for the first meeting, then it will be the last time slot), the other two to lectures. Examination Every fortnight (except the last) an assignment will be given  that has to be handed in (the grades for these 6 (or 5) assignments contribute 25% to the final grade). At the end of the 14 weeks students will be assigned a final project (suggestions are wellcome, the idea is to choose the topics in consultation  between students and lecturers). The students should take 7 to 8 days in a period of 5 weeks to produce a written elaboration, which contributes 50% to the final grade. The project, the earlier 6 assignments and, perhaps, a  designated part of the book/lecture material are discussed in an oral exam in week 6 after the end of the lectures. This oral exam might take the form of a presentation by the student about his/her project and it contributes the remaining 25% to the final grade. Literature Martin Schechter, Principles of Functional Analysis, AMS Graduate Studies in Mathematics Volume 36, 2nd edition 2001, ISBN 0-8218-2895-9 ; \$53,- ( \$ 47,- for members of the AMS) Remarks Course homepage http://www.math.leidenuniv.nl/~verduyn/colleges/FA2005
Last changed: 02-02-2016 12:27