Unit of study_

# MATH4313: Functional Analysis

### 2024 unit information

Functional analysis is one of the major areas of modern mathematics. It can be thought of as an infinite-dimensional generalisation of linear algebra and involves the study of various properties of linear continuous transformations on normed infinite-dimensional spaces. Functional analysis plays a fundamental role in the theory of differential equations, particularly partial differential equations, representation theory, and probability. In this unit you will cover topics that include normed vector spaces, completions and Banach spaces; linear operators and operator norms; Hilbert spaces and the Stone-Weierstrass theorem; uniform boundedness and the open mapping theorem; dual spaces and the Hahn-Banach theorem; and spectral theory of compact self-adjoint operators. A thorough mechanistic grounding in these topics will lead to the development of your compositional skills in the formulation of solutions to multifaceted problems. By completing this unit you will become proficient in using a set of standard tools that are foundational in modern mathematics and will be equipped to proceed to research projects in PDEs, applied dynamics, representation theory, probability, and ergodic theory.

## Unit details and rules

#### Mathematics and Statistics Academic Operations

Code MATH4313 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? None None None Real Analysis and abstract linear algebra (e.g., MATH2X23 and MATH2X22 or equivalent), and, preferably, knowledge of Metric Spaces

At the completion of this unit, you should be able to:

• LO1. demonstrate a coherent and advanced understanding of the key concepts of geometry of normed spaces, Hilbert Space Theory, Abstract Fourier Analysis, Hahn-Banach Theory and Spectral Theory, and how they provide a unified approach to infinite-dimensional linear problems in mathematics
• LO2. apply the fundamental ideas and results in functional analysis to solve given problems
• LO3. distinguish and compare the properties of different types of linear operators, analysing their spectra and deriving their main properties
• LO4. formulate analytic problems in functional-analytic terms and determine the appropriate framework to solve them
• LO5. communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work
• LO6. devise computational solutions to complex problems in functional analysis
• LO7. compose correct proofs of unfamiliar general results in functional analysis.

## Unit availability

This section lists the session, attendance modes and locations the unit is available in. There is a unit outline for each of the unit availabilities, which gives you information about the unit including assessment details and a schedule of weekly activities.

The outline is published 2 weeks before the first day of teaching. You can look at previous outlines for a guide to the details of a unit.

Session MoA   Location Outline
Semester 1 2024
Normal day Camperdown/Darlington, Sydney
Session MoA   Location Outline
Semester 1 2020
Normal day Camperdown/Darlington, Sydney
Semester 1 2021
Normal day Camperdown/Darlington, Sydney
Semester 1 2021
Normal day Remote
Semester 1 2022
Normal day Camperdown/Darlington, Sydney
Semester 1 2022
Normal day Remote
Semester 1 2023
Normal day Camperdown/Darlington, Sydney
Semester 1 2023
Normal day Remote

### Modes of attendance (MoA)

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