• Class Number 6233
• Term Code 3160
• Class Info
• Unit Value 6 units
• Mode of Delivery In Person
• COURSE CONVENER
• Dr Pierre Portal
• LECTURER
• Dr Pierre Portal
• Class Dates
• Class Start Date 26/07/2021
• Class End Date 29/10/2021
• Census Date 14/09/2021
• Last Date to Enrol 02/08/2021
SELT Survey Results

Advanced Functional Analysis, Spectral theory and Applications (MATH3325)

This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered include:

Hilbert spaces - bounded linear operators, compact operators, the spectral theorem for compact self-adjoint operators; Fourier transform, applications to partial differential equations and the central limit theorem.

Measure theory - abstract measure theory, integration,  Fubini-Tonelli theorem, Radon-Nikodym theorem, Hausdorff measure, fractals.

Banach spaces and linear operators - basic properties, Baire category theorem and its consequences (uniform boundedness principle, closed graph and open mapping theorems), Hahn-Banach theorem and dual spaces, sequential version of Banach-Alaoglu theorem, dual spaces of L^p spaces and spaces of continuous functions. Applications to Fourier series, fractals.

Note: This is an HPC. It emphasises mathematical rigour and proof and continues the development of modern analysis from an abstract viewpoint.

## Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of functional analysis and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of functional analysis techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from functional analysis
4. Apply problem-solving using functional analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts

## Research-Led Teaching

The essay component of the course is an opportunity for students to develop research skills, including the ability to digest the dense content of research papers.

Functional Analysis, Sobolev Spaces and Partial Differential Equations by H. Brezis.

Functional Analysis by E. Stein and R. Shakarchi.

Recommended student system requirements

ANU courses commonly use a number of online resources and activities including:

• video material, similar to YouTube, for lectures and other instruction
• two-way video conferencing for interactive learning
• email and other messaging tools for communication
• interactive web apps for formative and collaborative activities
• print and photo/scan for handwritten work
• home-based assessment.

To fully participate in ANU learning, students need:

• A computer or laptop. Mobile devices may work well but in some situations a computer/laptop may be more appropriate.
• Webcam
• Speakers and a microphone (e.g. headset)
• Reliable, stable internet connection. Broadband recommended. If using a mobile network or wi-fi then check performance is adequate.
• Suitable location with minimal interruptions and adequate privacy for classes and assessments.
• Printing, and photo/scanning equipment

## Staff Feedback

Students will be given feedback in the following forms in this course:
• Feedback to the whole class, to groups, to individuals, focus groups

## Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

## Class Schedule

Week/Session Summary of Activities Assessment
1 Sobolev spaces, minimisation in Hilbert spaces
2 Spectral theorem for compact self-adjoint operators
3 Banach spaces, Hahn-Banach theorem
4 L^p spaces: duality and interpolation
5 Baire theorem and its famous applications (uniform boundedness, open mapping, closed graph) Assignment 1 due
6 Weak and weak* topologies, Banach-Alaoglu's theorem
7 Reflexive spaces, uniform convexity, and applications (projections on convex sets, weak* compactness in Sobolev spaces) Assignment 2 due
8 Schauder bases, characterisation of reflexivity
9 Unconditional bases, the Haar basis
10 Compact operators, Fredholm theory Assignment 3 due
11 Mean ergodic theorems
12 Mixing linear dynamical systems, Blum-Hanson theorem

## Assessment Summary

Assessment task Value Due Date Return of assessment Learning Outcomes
Assignment 1 10 % 24/08/2021 31/08/2021 1,2,3,4
Assignment 2 10 % 21/09/2021 28/09/2021 1,2,3,4
Assignment 3 10 % 12/10/2021 19/10/2021 1,2,3,4
Essay 30 % 12/11/2021 19/11/2021 1,2,3,4
Exam (take home) 40 % 04/11/2021 02/12/2021 1,2,3,4

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

## Policies

ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

## Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the ANU Online website Students may choose not to submit assessment items through Turnitin. In this instance you will be required to submit, alongside the assessment item itself, hard copies of all references included in the assessment item.

## Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

## Examination(s)

The date range indicates the start of the end of semester exam period and the date official end of semester results are released on ISIS. Please check the course Wattle site and the ANU final Examination Timetable http://www.anu.edu.au/students/program-administration/assessments-exams/examination-timetable to confirm the date, time and location of the exam.

Value: 10 %
Due Date: 24/08/2021
Return of Assessment: 31/08/2021
Learning Outcomes: 1,2,3,4

Assignment 1

Prove several statements similar to the statements discussed in lectures and workshops. Hand in a complete typed or scanned personal solution.

Value: 10 %
Due Date: 21/09/2021
Return of Assessment: 28/09/2021
Learning Outcomes: 1,2,3,4

Assignment 2

Prove several statements similar to the statements discussed in lectures and workshops. Hand in a complete typed or scanned personal solution.

Value: 10 %
Due Date: 12/10/2021
Return of Assessment: 19/10/2021
Learning Outcomes: 1,2,3,4

Assignment 3

Prove several statements similar to the statements discussed in lectures and workshops. Hand in a complete typed or scanned personal solution.

Value: 30 %
Due Date: 12/11/2021
Return of Assessment: 19/11/2021
Learning Outcomes: 1,2,3,4

Essay

Write a 5 page summary of the advanced/research material you have read. The topic should be decided jointly with the lecturer.

Value: 40 %
Due Date: 04/11/2021
Return of Assessment: 02/12/2021
Learning Outcomes: 1,2,3,4

Exam (take home)

In 3 hours, prove several statements similar to the statements discussed in lectures and workshops. Hand in a complete scanned personal solution.

The date range indicates the start of the end of semester exam period and the date official end of semester results are released on ISIS. Please check the course Wattle site and the ANU final Examination Timetable http://www.anu.edu.au/students/program-administration/assessments-exams/examination-timetable to confirm the date, time and location of the exam.

## Online Submission

The ANU uses Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. While the use of Turnitin is not mandatory, the ANU highly recommends Turnitin is used by both teaching staff and students. For additional information regarding Turnitin please visit the ANU Online website.

## Hardcopy Submission

For some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.

## Late Submission

Late submission of assessment tasks without an extension are penalised at the rate of 5% of the possible marks available per working day or part thereof. Late submission of assessment tasks is not accepted after 10 working days after the due date, or on or after the date specified in the course outline for the return of the assessment item.

## Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

## Returning Assignments

Marked pieces of assessment are returned to the student via the Wattle site.

## Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure The Course Convener may grant extensions for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

## Resubmission of Assignments

Resubmission of assignments is possible up to the submission deadline, but not after.

## Privacy Notice

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes. Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

## Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

## Convener

 Dr Pierre Portal 61253384 Pierre.Portal@anu.edu.au

### Research Interests

Harmonic, stochastic, and functional analysis. Deterministic and stochastic PDE.

### Dr Pierre Portal

 By Appointment By Appointment

## Instructor

 Dr Pierre Portal 61253384 pierre.portal@anu.edu.au

### Dr Pierre Portal

 By Appointment By Appointment