Overview

Description

CALENDAR DESCRIPTION:

Infinite dimensional vector spaces, convergence, generalized Fourier series. Operator Theory; the Fredholm alternative. Application to integral equations and Sturm-Liouville systems. Spectral theory.

COURSE DETAILS:

Course description:
This will be an introduction to functional analysis and some of its applications. In a nutshell, we'll investigate the properties of continuous linear mappings of infinite-dimensional vector spaces. The more structure the space has, the stronger the theorems. We will focus on Hilbert spaces (which are almost 'tame') and Banach spaces (which can be wilder). The results will provide extremely useful tools for applications ranging from fixed point theory to numerical approximation through to the solvability of partial differential equations. We could also sell this course as the 'fundamental underpinning of much of machine learning', but we won't.

Topics covered: We'll quickly review important definitions concerning metric spaces and normed spaces, especially the notions of completeness and compactness. We'll then introduce Banach spaces and bounded linear operators on them. We shall then discuss Hilbert spaces and their duals, some applications, and then onto spectral theory/Sturm-Liouville theory. We will pause to marvel at the simple yet far-reaching Riesz Representation theorem and the Lax-Milgram Lemma. We then return to abstract Banach spaces, and the 'big' theorems: the Hahn-Banach theorem, the Uniform Boundedness theorem, the Open Mapping theorem and the Closed Graph theorem. The course ends with a return to spectral theory, and the Fredholm Alternative for compact operators.

Structure and Assessment: There will be 4 hours of lecture, and 1 hour of problem session (for a total of 5 hours of contact time) per week. There will be 12 equally-weighted weekly assignments. Students will present their solutions in the problem sessions, have an opportunity to correct their own work, and the assessment will be based on the oral and written solutions. Note that we may, based on the success of these problem sessions, change the method of assessment to include a final examinatiom.

The work you hand in must represent your own efforts. Plagiarism is a serious violation of SFU'S Academic Honesty and Student Conduct policy, and we maintain a zero-tolerance approach to such violations.

Grading

Materials

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