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**Dr Denis PotapovThe University of New South Wales**

#### Synopsis

Functional analysis a central pillar of modern analysis and its foundations will be covered in this course with an emphasis on the study of bounded linear maps between topological linear spaces. This provides the basic tools for the development of such areas as quantum mechanics, harmonic analysis and stochastic calculus.

Course content:

- Review of general Functional Analysis (measure and integration, normed linear spaces, dual spaces, linear operators, function spaces, Hilbert spaces).
- Fourier analysis
- Spectral theory of operators (spectral theorem and functional calculus for finite-dimensional, bounded and unbounded linear operators)
- Applications: mathematical model of quantum mechanics, modern noncommutative analysis (if time permits).

#### Contact hours

28 hours of hours spread over four weeks, plus consultation as required.

#### Prerequisites

- Basics of set theory (countable and uncountable sets, Schroeder-Bernstein Theorem)
- Topological and metric spaces (convergence, open and closed sets, continuity, completeness, contraction mapping theorem)
- Sequences and series of functions on metric spaces (pointwise and uniform convergence, differentiation and integration of limits and sums).

#### Resources

- J.B. Conway: A Course in Functional Analysis.
- W. Rudin: Functional Analysis.
- M. Reed and B. Simon: Methods of Modern Mathematical Physics. Vol. 1 Functional Analysis.
- K. Yosida: Functional Analysis

#### About Denis Potapov

Research interests: Noncommutative analysis and noncommutative integration; perturbation theory; noncommutative and vector-valued harmonic analysis.