MATH5605 is a Honours and Postgraduate coursework Mathematics course. It is a core course for all Pure Mathematics Honours students. See the course overview below.

**Units of credit:** 6

**Prerequisites:** MATH3611

**Cycle of offering:** Term 1 in Trimester.

**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 contains information about the course. (The timetable is only up-to-date if the course is being offered this year.)

If you are currently enrolled in MATH5605, you can log into UNSW Moodle for this course.

#### 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 provides 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 spaces, 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.