Cornell Math - MATH 622 Spring 2007

MATH 622: Functional Analysis with Applications (Spring 2007)

Instructor: Mohammad Asadzadeh

Meeting Time & Room

Part I: Introductory material. The goal, in this part, is to give an introduction to functional analysis as a fundamental tool in some central areas of mathematics and applied mathematics as, e.g., ordinary and partial differential equations, mathematical statistics and numerical analysis.

We cover introductory material viz, Normed vector spaces. Banach- and Hilbert spaces. An orientation about Lebesgue integral. Contractive mappings. Fix-point theorems. Compactness. Operators in Hilbert spaces. Spectral theory for the compact, self-adjoint operators.Fredholm alternative theorem. Applications to differential- and integral equations. Optimization problem.

Part II: A somewhat advanced approach. In this part the basic idea is to apply geometric methods to functions and function spaces. A function is considered as a point in a space, and this space will be a vector space of infinite dimension. Geometrical objects like balls, and also convergence, are introduced in these spaces.

The course will consists of lectures and exercises sessions based on [1], [2], and hand outs during the course. The balance between the 2 parts will depend upon the background of the audience according to C = α(I) + β(II), 0 < α, β < 1.

Prerequisites. Basic linear algebra, basic analysis in one and several dimensions.

Rough plan for part II

  • Normed linear spaces
  • Lp spaces and their duals
  • The Hahn-Banach theorem
  • Baire's theorem with consequences
  • Hilbert spaces
  • The separable case of Alaoglu's theorem
  • Topological spaces, Hausdorff spaces, compactness, Tychnonv's theorem
  • Topological vector spaces, weak topologies, the general case of Alaoglu's theorem
  • Urysohn's lemma in locally Hausdorff spaces
  • The Riesz representation theorem, positive case.

References

[1] Debnath L. and Mikusinski, P. Introduction to Hilbert Spaceswith Applications, 2nd ed., Academic Press, 1999.

[2] Folland, G. Real Analysis. Modern Techniques and their Applications, John Wiley & Sons, 1999 (Chapters 5-7 and parts of Chapter 4).