This course is intended both for continuing mathematics students 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:

• Measure theory
  • Functions of bounded variation over R
  • Absolute continuity and integration
  • Examples of more general measures (Radon, Hausdorff, probability measures)
  • Fubini-Tonelli theorem
  • Radon-Nikodym theorem
• Banach spaces and linear operators
  • Classical function and sequence spaces
  • Hahn-Banach theorem
  • Closed graph and open mapping theorems
  • Uniform boundedness principles
  • Sequential version of Banach-Alaoglu theorem
  • Spectrum of an operator and analysis of the compact self-adjoint case
  • Fredholm alternative theorem.

Note: Graduate students attend joint classes with undergraduates but will be assessed separately.

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

Materials from current research will be incorporated into the class as appropriate.

Recommended Resources

Real Analysis by E. Stein and R. Shakarchi.

Functional Analysis by E. Stein and R. Shakarchi.

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

There are a variety of online platforms you will use to participate in your study program at ANU as a whole. These could include videos 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/or photo/scan for handwritten work and drawings, and home-based assessment.

ANU outlines recommended student system requirements to ensure you are able to participate fully in your learning. Other information is also available about the various Learning Platforms you may use.

Staff Feedback

Students will be given feedback in the following forms in this course:

• Written comments
• Verbal comments
• 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	Hilbert spaces: examples, orthogonality, linear functionals	Lectures on Tuesday, workshops on Friday
2	Hilbert spaces: bounded linear operators, projections, compact operators
3	Hilbert spaces: spectral theorem for compact operators	Homework 1 due
4	Banach spaces: L^p spaces, duality
5	Banach spaces: Hahn-Banach theorem
6	Baire Category theorem	Homework 2 due
7	Uniform boundedness theorem	Mid-term assessment
8	Open mapping theorem, closed graph theorem
9	Application of L^p spaces	Homework 3 due
10	abstract measure theory, spectral theorem for bounded symmetric operators on Hilbert spaces
11	weak and weak* topologies, Banach-Alaoglu theorem
12	compact operators, Fredholm theory	Homework 4 due

Tutorial Registration

Please see MyTimetable and Wattle for details: there will be one workshop session each week (Friday), in which students are required to be enrolled.

Assessment Summary

Assessment task	Value	Learning Outcomes
In class quizzes	15 %	1,2,3,4
Homework	15 %	1,2,3,4
In class mid-semester assessment	25 %	1,2,3,4
Final exam	45 %	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:

• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Policy and General Information
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure

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.

Participation

The course will operate with a 2-hour lecture on Tuesdays, where the results of the course will be presented and proved. On Fridays, we will have a mix of informal lectures and workshops, where there will be lots of discussions of the course materials.

Important note: all announcements will be made in class. It is expected that students will attend all lectures and workshops where possible.

Examination(s)

The dates and times of the mid-semester assessment and the final examination will be announced in class.

Assessment Task 1

In class quizzes

10-15 minute quizzes at the end of Tuesday lectures. Every week except weeks 1 and 7. The quiz will consist of one or two questions that will test your knowledge and direct understanding of the theorems from the course.

There are no make-up quizzes, but the 2 lowest scores will be dropped at the end of the semester. If you have a reason that made you miss the class 3 times or more, please provide appropriate justification to the convener.

Assessment Task 2

Homework

You will be required to prove several statements similar to the statements discussed in lectures and workshops and submit a complete typed or scanned personal solutions. There will be four homework assignments throughout the semester. Submission of assignments is in PDF file format, via Wattle. Any PDF file has to be easily printable in black & white on A4 paper, and its contents have to be clearly readable after printing. Students enrolled with MATH6214 will have additional questions for the homework assignments compared to those enrolled in MATH3325.

Assessment Task 3

In class mid-semester assessment

This is a summative assessment. You will be required to solve problems using knowledge from the first half of the course and submit personal solutions.

Assessment Task 4

Final exam

This is a summative assessment. You will be required to solve problems using knowledge from the whole course and submit personal solutions.

Academic Integrity

Academic integrity is a core part of our culture as a community of scholars. At its heart, academic integrity is about behaving ethically. This means that all members of the community commit to honest and responsible scholarly practice and to upholding these values with respect and fairness. The Australian National University commits to embedding the values of academic integrity in our teaching and learning. We ensure that all members of our community understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with. The University has policies and procedures in place to promote academic integrity and manage academic misconduct. Visit the following Academic honesty & plagiarism website for more information about academic integrity and what the ANU considers academic misconduct. The ANU offers a number of services to assist students with their assignments, examinations, and other learning activities. The Academic Skills and Learning Centre offers a number of workshops and seminars that you may find useful for your studies.

Online Submission

Submission of assignments is in PDF file format, via Wattle. Any PDF file has to be easily printable in black & white on A4 paper, and its contents have to be clearly readable after printing.

MATH3325 does not use Turnitin, having been granted an exemption.

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. Quizzes, the in-class mid-semester assessment, and the final exam cannot be submitted late. Late submission of other assessment tasks is not accepted after 5 working days after the due date, or on or after the date specified in the course outline for the return of the assessment item, or after solutions to an assessment task have been released (whichever occurs first).

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 course 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

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information. In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service — including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy. If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

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).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents undergraduate and ANU College students
• PARSA supports and represents postgraduate and research students