Unit of study_

# MATH4313: Functional Analysis

## Overview

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.

### Details

Academic unit Mathematics and Statistics Academic Operations MATH4313 Functional Analysis Semester 1, 2023 Normal day Camperdown/Darlington, Sydney 6

### Enrolment rules

 Prohibitions ? None None None Real Analysis and abstract linear algebra (e.g., MATH2X23 and MATH2X22 or equivalent), and, preferably, knowledge of Metric Spaces Yes

### Teaching staff and contact details

Coordinator James Parkinson, james.parkinson@sydney.edu.au Daniel Daners James Parkinson

## Assessment

Type Description Weight Due Length
Supervised exam

Final Exam
Written mathematical formulae and arguments.
55% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
Written assignment
15% Week 04
Due date: 15 Mar 2023 at 23:59

Closing date: 22 Mar 2023
14 days
Outcomes assessed:
Assignment Assignment 2
Written assignment
15% Week 08
Due date: 19 Apr 2023 at 23:59

Closing date: 26 Apr 2023
14 days
Outcomes assessed:
Assignment Assignment 3
Written assignment
15% Week 12
Due date: 17 May 2023 at 23:59

Closing date: 24 May 2023
14 days
Outcomes assessed:
• Writing task: 3 assignments worth 10% each. These assignments will require you to synthesise information from lectures and tutorials to create concise written arguments.
• Final exam: The exam will cover all material in the unit from both lectures and  tutorials.
• Final Exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator.

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

At HD level, a student demonstrates a flair for the subject as well as a detailed and comprehensive understanding of the unit material. A ‘High Distinction’ reflects exceptional achievement and is awarded to a student who demonstrates the ability to apply their subject knowledge and understanding to produce original solutions for novel or highly complex problems and/or comprehensive critical discussions of theoretical concepts.

Distinction

75 - 84

At DI level, a student demonstrates an aptitude for the subject and a well-developed understanding of the unit material. A ‘Distinction’ reflects excellent achievement and is awarded to a student who demonstrates an ability to apply their subject knowledge and understanding of the subject to produce good solutions for challenging problems and/or a reasonably well-developed critical analysis of theoretical concepts.

Credit

65 - 74

At CR level, a student demonstrates a good command and knowledge of the unit material. A ‘Credit’ reflects solid achievement and is awarded to a student who has a broad general understanding of the unit material and can solve routine problems and/or identify and superficially discuss theoretical concepts.

Pass

50 - 64

At PS level, a student demonstrates proficiency in the unit material. A ‘Pass’ reflects satisfactory achievement and is awarded to a student who has threshold knowledge.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

### Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Review of the topology of metric spaces, compactness and continuity Lecture (3 hr)
Week 02 Normed linear spaces and basic properties of linear operators Lecture and tutorial (4 hr)
Week 03 Finite and infinite dimensional Banach spaces Lecture and tutorial (4 hr)
Week 04 Banach algebras and the Stone-Weierstrass Theorem Lecture and tutorial (4 hr)
Week 05 Hilbert spaces, orthonormal systems, projections and abstract Fourier series Lecture and tutorial (4 hr)
Week 06 Baire's theorem and the open mapping theorem Lecture and tutorial (4 hr)
Week 07 The closed graph theorem, the uniform boundedness theorem Lecture and tutorial (4 hr)
Week 08 Closed operators Lecture and tutorial (4 hr)
Week 09 Duality and the Hahn-Banach theorem Lecture and tutorial (4 hr)
Week 10 Weak convergence Lecture and tutorial (4 hr)
Week 11 Duality in Hilbert spaces and the Lax-Milgram Theorem Lecture and tutorial (4 hr)
Week 12 Basics of spectral theory, ascent and descent of linear operators Lecture and tutorial (4 hr)
Week 13 The spectrum of compact operators Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Daniel Daners, Introduction to Functional Analysis, University of Sydney, 2021 (available on Canvas)

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

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.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.