UTSA logo Functional Analysis
MAT 5403
Spring 2003
Instructor: J. Iovino
 

The course has been scheduled for M,W, between 3:30pm and 4:45pm, at HSS 3.03.10 (1604 campus).

Brief Description

Functional analysis is one of the pillars of classical mathematics, and an indispensable language in any of the fields of mathematics which are based on analysis, e.g., differential equations, harmonic analysis, analytic number theory, numerical analysis, optimization, mathematical physics, and probability theory. It is also the place where several areas of mathematics, such as real analysis, linear algebra, probability and topology, meet.

The course will provide an introduction to the fundamental concepts of linear analysis: Banach spaces, Hilbert spaces, operators, and duality.

Prerequisites

In addition to linear algebra, the course assumes knowledge of basic topology and Lebesgue measure theory, as included in the course Theory of Functions of a Real Variable I (MAT 5203).

Content

  • Basic Concepts. Banach spaces, subspaces, quotient spaces, the Holder and Minkowski inequalities, the classical Banach spaces, Hilbert spaces lp, c0, Lp[0,1], C[0,1], Riesz' lemma, separability, orthonormal bases in Hilbert spaces.
  • The Hahn-Banach Theorem, Dual Spaces. Banach spaces, subspaces, quotient spaces, the Holder and Minkowski inequalities, the classical Banach spaces, Hilbert spaces, Riesz' lemma, separability, orthonormal bases in Hilbert spaces.
  • Weak Topologies. The weak topology, the weak-star topology, the second dual, bounded sets, the Banach-Steinhaus Theorem, the Krein-Milman Theorem, the James-Simmonds-Godefroy theorem on James boundaries of sets, James' characterization of weak compactness, the Eberlein-Smullyan and Krein-Smullyan Theorems.
  • The Open Mapping Theorem. Banach's Open Mapping and Closed Graph theorems, complementability, Sobczyk's theorem on complementability of c0, Phillip's theorem on the noncomplementability of c0 in l∞. Every separable Banach space can occurs as a subspace of C[0,1].
  • Schauder Bases. The notion of Schauder basis, James' theorem on the characterization of reflexivity in terms of bases, Mazur's basic sequence theorem, the Krein-Milman-Rutman stability result, Pelczynski's theorem on subspaces of lp-spaces, Unconditional Schauder bases, James' theorem on the containment of lp and c0.

    • Textbook

    Lecture notes will be distributed and no textbook will be required. However, the following book is recommended as reference:

    Fabian, M.; Habala, P.; Hajek, P.; Montesinos Santalucia, V.; Pelant, J.; Zizler, V, Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics. Volume. 8, Springer-Verlag, 2001.

    Evaluation

    There will be four problem sets of five exercises each. The students will have two weeks to work on each set. Each problem set will be worth 25% of the grade.

    Schedule

    For an outline of the material covered each day of the academic semester, click here.

    How to contact the instructor

    Office: SB 4.01.34 (Directions: Go to the fourth floor of the Science Building and as you get off the elevator follow the arrows to the Applied Mathematics Department. My office is right across the hall from the main office.)

    Telephone: (210) 458-5531

    Email: iovino@math.utsa.edu

    Office hours: M,W 10:11:30 am, or by appointment.

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