MATHS 731 : Functional Analysis


2020 Semester Two (1205) (15 POINTS)

Course Prescription

Provides the mathematical foundations behind some of the techniques used in applied mathematics and mathematical physics; it explores how many phenomena in physics can be described by the solution of a partial differential equation, for example the heat equation, the wave equation and Schrödinger's equation. Recommended preparation: MATHS 730 and 750.

Course Overview

This course builds upon the topics from MATHS 332 and MATHS 333, as well as, the topics in MATHS 730 and MATHS 750. It provides the basis of modern Analysis, as developed in the 20th Century. Anyone contemplating a career in Analysis/Functional Analysis should complete this course. In terms of specific content the course covers:  Finite Dimensional Banach spaces; Hilbert Spaces; The Hahn-Banach Theorem (including Separation Theorems); Open Mapping Theorem; Closed Graph Theorem and Uniform Boundedness Theorem.  The course also covers aspects of  Banach Algebras, including the Gelfand Transform and also covers the Spectral Mapping Theorem for compact normal operator  defined on Hilbert spaces.

Course Requirements

Prerequisite: MATHS 332 and 333

Capabilities Developed in this Course

Capability 1: Disciplinary Knowledge and Practice
Capability 2: Critical Thinking
Capability 3: Solution Seeking

Learning Outcomes

By the end of this course, students will be able to:
  1. Understand the fundamentals of functional analysis (Capability 1)
  2. Apply the fundamental tools of functional analysis to solve some problems in analysis (Capability 2 and 3)


Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Final Exam 70% Individual Coursework
Assessment Type Learning Outcome Addressed
1 2
Final Exam

The final grade will be either 30% Assignments marks, and 70% Exam or 100% exam, which ever favours the student.

Key Topics

Banach spaces and Banach Algebras

Learning Resources

Course book

Special Requirements

Must obtain at least 35% in the final exam.

Workload Expectations

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 36 hours of lectures, 72 hours of reading and thinking about the content and 12 hour of work on assignments and/or test preparation.

Digital Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.


The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website at

Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page:

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter (


Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 11/01/2020 03:12 p.m.