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MATH7901 – Analysis

2026 – Session 1, In person-scheduled-weekday, North Ryde

General Information

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Unit convenor and teaching staff Unit convenor and teaching staff
Xuan Duong
xuan.duong@mq.edu.au
Ji Li
ji.li@mq.edu.au
Credit points Credit points
10
Prerequisites Prerequisites
Admission to GradDipRes or GradCertRes
Corequisites Corequisites
Co-badged status Co-badged status
Unit description Unit description
This unit provides an advanced introduction to the key areas of research interest in modern analysis. We will study Lebesgue integration, positive Borel measures, and the all important function spaces Lp. Then we will study the elementary Hilbert space theory and Banach space techniques. This will provide familiarity with some of the major theorems which make up the analysis toolbox: Monotone and Dominated Convergence theorems; Fatou's lemma; Egorov's theorem; Lusin's theorem; Radon-Nikodym theorem; Fubini-Tonelli theorems about product measures and integration on product spaces; Uniform Boundedness; Fundamental Theorem of Calculus for Lebesgue Integrals; Minkowski's Inequality; Holder's Inequality; Jensen's Inequality; and Bessel's Inequality.

Important Academic Dates

Information about important academic dates including deadlines for withdrawing from units are available at https://www.mq.edu.au/study/calendar-of-dates

Learning Outcomes

On successful completion of this unit, you will be able to:

  • ULO1: Demonstrate the abstract setting and the logical arguments in metric spaces, Banach spaces and Hilbert spaces, and formulate the structure on positive Borel measure, and the Lebesgue spaces, including the Riesz representation theorem and the approximation by continuous functions
  • ULO2: Analyse the abstract setting of metric space, Banach space and Hilbert space, and  evaluate their utility to study particular areas of differential equations and calculus in higher dimension setting.
  • ULO3: Analyse the structure of Lebesgue spaces, and evaluate the utility of this setting to study particular areas of Fourier series and Fourier transforms, harmonic functions and Poisson integral
  • ULO4: Create a precise mathematical problem from the specific modelling by applying these abstract settings, and discuss the solution by synthesising the methods in the theory of Banach space and Hilbert space

General Assessment Information

ATTENDANCE and PARTICIPATION: Please contact the unit convenor as soon as possible if you have difficulty attending and participating in any classes. There may be alternatives available to make up the work. If there are circumstances that mean you will miss a class, you can apply for Special Consideration via ask.mq.edu.au.

ASSIGNMENT SUBMISSION: Assignment submission will be online through the iLearn page.

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You may submit as often as required prior to the due date/time. Please note that each submission will completely replace any previous submissions. It is in your interests to make frequent submissions of your partially completed work as insurance against technical or other problems near the submission deadline.

 

Assessment Tasks

Name Weighting Hurdle Due Groupwork/Individual Short Extension AI Approach
Project 30% No 07/06/2026 Individual Yes Open
Assignment 50% No 24/05/2026 Individual Yes Open
Presentation 20% No Exam Period Individual No Observed

Project

Assessment Type 1: Experiential task
Indicative Time on Task 2: 18 hours
Due: 07/06/2026
Weighting: 30%
Groupwork/Individual: Individual
Short extension 3: Yes
AI Approach: Open

This project gives you the opportunity to apply the knowledge gained in the unit to a larger problem than the short questions typical in assignments.
On successful completion you will be able to:
  • Demonstrate the abstract setting and the logical arguments in metric spaces, Banach spaces and Hilbert spaces, and formulate the structure on positive Borel measure, and the Lebesgue spaces, including the Riesz representation theorem and the approximation by continuous functions
  • Analyse the abstract setting of metric space, Banach space and Hilbert space, and  evaluate their utility to study particular areas of differential equations and calculus in higher dimension setting.
  • Analyse the structure of Lebesgue spaces, and evaluate the utility of this setting to study particular areas of Fourier series and Fourier transforms, harmonic functions and Poisson integral
  • Create a precise mathematical problem from the specific modelling by applying these abstract settings, and discuss the solution by synthesising the methods in the theory of Banach space and Hilbert space

Assignment

Assessment Type 1: Problem-based task
Indicative Time on Task 2: 30 hours
Due: 24/05/2026
Weighting: 50%
Groupwork/Individual: Individual
Short extension 3: Yes
AI Approach: Open

The assignment will test your ability to solve mathematical problems using concepts and techniques learnt in the unit.
On successful completion you will be able to:
  • Demonstrate the abstract setting and the logical arguments in metric spaces, Banach spaces and Hilbert spaces, and formulate the structure on positive Borel measure, and the Lebesgue spaces, including the Riesz representation theorem and the approximation by continuous functions
  • Analyse the abstract setting of metric space, Banach space and Hilbert space, and  evaluate their utility to study particular areas of differential equations and calculus in higher dimension setting.
  • Analyse the structure of Lebesgue spaces, and evaluate the utility of this setting to study particular areas of Fourier series and Fourier transforms, harmonic functions and Poisson integral
  • Create a precise mathematical problem from the specific modelling by applying these abstract settings, and discuss the solution by synthesising the methods in the theory of Banach space and Hilbert space

Presentation

Assessment Type 1: Presentation task
Indicative Time on Task 2: 12 hours
Due: Exam Period
Weighting: 20%
Groupwork/Individual: Individual
Short extension 3: No
AI Approach: Observed

You will present a talk based on your project.
On successful completion you will be able to:
  • Demonstrate the abstract setting and the logical arguments in metric spaces, Banach spaces and Hilbert spaces, and formulate the structure on positive Borel measure, and the Lebesgue spaces, including the Riesz representation theorem and the approximation by continuous functions
  • Analyse the abstract setting of metric space, Banach space and Hilbert space, and  evaluate their utility to study particular areas of differential equations and calculus in higher dimension setting.
  • Analyse the structure of Lebesgue spaces, and evaluate the utility of this setting to study particular areas of Fourier series and Fourier transforms, harmonic functions and Poisson integral
  • Create a precise mathematical problem from the specific modelling by applying these abstract settings, and discuss the solution by synthesising the methods in the theory of Banach space and Hilbert space

1 If you need help with your assignment, please contact:

  • the academic teaching staff in your unit for guidance in understanding or completing this type of assessment
  • Academic Success for academic skills support.

2 Indicative time-on-task is an estimate of the time required for completion of the assessment task and is subject to individual variation.

3 An automatic short extension is available for some assessments. Apply through the Service Connect Portal.

Delivery and Resources

Related materials:  Real Analysis, Lebesgue Integrals, Linear functional analysis

Unit Schedule

Week 1: Revision on metric spaces and convergence

 

Week 2: Compactness in metric spaces

 

Week 3: Banach spaces and convergence of sequences

 

Week 4: Lebesgue Outer Measure & Measurable Sets

 

Week 5: Measurable Functions

 

Week 6: Measurable Functions and The Lebesgue Integral (Construction)

 

Week 7: The Convergence Theorems (Part I)

 

Week 8: The Convergence Theorems (Part II) and L^1 space

 

Week 9: Product Measures and the Fubini--Tonelli Theorems

 

Week 10: Elementary Hilbert space theory 

 

Week 11: Hilbert spaces: Trigonometric series

 

Week 12: Applications 

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Academic Integrity

At Macquarie, we believe academic integrity – honesty, respect, trust, responsibility, fairness and courage – is at the core of learning, teaching and research. We recognise that meeting the expectations required to complete your assessments can be challenging. So, we offer you a range of resources and services to help you reach your potential, including free online writing and maths support, academic skills development and wellbeing consultations.

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Unit information based on version 2026.02 of the Handbook