Math 617 Functional Analysis

Spring 2020, UAF

Ed Bueler course details:
Chapman 306C MWF 10:30--11:30am
elbueler@alaska.edu Bunnell 410
bueler.github.io crn:  35419
2D Gibbs phenomenon for Fourier series of a square wave

Functional analysis is the theory of infinite-dimensional vector spaces and linear maps upon them. It is the mathematical home of field theories (electromagnetic and gravitational), fluid mechanics, signal processing, and especially quantum mechanics.

bound states (eigenfunctions) of the harmonic oscillator

The mathematical topics are the study of Banach and Hilbert spaces, continuous linear maps (operators) on them, and choices of different topologies. Major results include the Baire Category theorem, the Hahn-Banach theorem, the Riesz representation theorem (for measures), the open mapping theorem, the closed graph theorem, the spectral theorem, and the functional calculus. Compact, bounded, and self adjoint operators will be considered, especially their spectral properties.

positive/negative zones for bound states of the hydrogen atom

Mathematically-inclined students from the sciences and engineering are encouraged to register, as are graduate students in mathematics looking for an elective with practical relevance. This course is particularly aimed at students interested in the mathematics of quantum mechanics.

The official prerequisites are these:  MATH F314 Linear Algebra and MATH 401 Introduction to Real Analysis, or permission of instructor. Recommended: MATH F422 Introduction to Complex Analysis and MATH F641 Real Analysis.

For graduate students with a background from another university, these prerequisites imply some rigorous, though introductory, exposure to the analysis of real functions, plus exposure to linear algebra with a little rigor, plus the basics of complex numbers. The difficulty of assigned work will depend on the student's comfort doing proofs with real functions and vectors.

Required textbook: J. Muscat, Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, Springer, 2014

Strongly-recommended textbook: B. Hall, Quantum Theory for Mathematicians, Springer, 2013.