School of Mathematics and Statistics

MATH5605 Functional Analysis

MATH5605 is a Mathematics Level V course. See the course overview below.

Units of credit: 6

Prerequisites: MATH3611

Cycle of offering: Course not offered every year - see School for more details.

Graduate attributes: the course will enhance your research, inquiry and analytical thinking abilities.

More information: this recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains up-to-date timetabling information.

The following links provide a Set of Problem Sheets and a Set of Skeleton Notes

Course Overview

This course can be thought of as a continuation of Higher Analysis MATH3611. Functional analysis a central pillar of modern analysis, and we will cover its foundations.
The main emphasis will be on the study of the properties of bounded linear maps between topological linear spaces of various kinds. This provide the basic tools for the development of such areas as quantum mechanics, harmonic analysis and stochastic calculus. It also has a very close relation to measure and integration theory (MATH5825).

Detailed course schedule

Normed linear spaes, bounded operators, Banach spaces.
Functionals and Hahn-Banach theorems.
The Baire-category theorem, the principle of uniform boundedness, the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem.
Hilbert space theory; orthonormality, the Riesz representation theorem, projections, convexity.
Operators on Hilbert spaces, normal and selfadjoint operators, spectrum and resolvent, Spectral mapping theorem.
Compact operators, their spectral data, the spectral theorem.
(If time permits) Nonlinear functional analysis: Frechet differential, Taylor series, analytic functions on Banach space.