Real Analysis: Advanced (MAST20033)
Undergraduate level 2Points: 12.5On Campus (Parkville)
Overview
| Availability | Semester 1 |
|---|---|
| Fees | Look up fees |
This subject introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. Many of the important results are proved rigorously and students are introduced to methods of proof such as mathematical induction and proof by contradiction.
The important distinction between the real numbers and the rational numbers is emphasised and used to motivate rigorous notions of convergence and divergence of sequences, including the Cauchy criterion. Various constructions of the real numbers, for example using Dedekind cuts or by completion, are discussed and shown to be equivalent. These ideas are extended to cover the theory of infinite series, including common tests for convergence and divergence. Compactness of the unit interval is established and various consequences of compactness, such as the Extreme Value Theorem, are discussed. A similar treatment of continuity and differentiability of functions of a single variable leads to applications such as the Mean Value Theorem and Taylor’s theorem. We define the Riemann integral and explore its properties, and we prove the Fundamental Theorem of Calculus. The convergence properties of sequences and series are explored, with applications to power series representations of elementary functions and their generation by Taylor series. Fourier series are introduced as a way to represent periodic functions. Further topics may include: uniform continuity, equicontinuity, the Arzela-Ascoli theorem, and the Stone-Weierstrass theorem.
Intended learning outcomes
On completion of this subject, students should be able to:
- Apply rigour in mathematics, be able to use proof by induction, proof by contradiction, and to use epsilon-delta proofs both as a theoretical tool and a tool of approximation;
- Describe the theory and applications of both the Riemann and Lebesgue integrals;
- Identify real numbers and explain their importance of completeness;
- Define convergence and the Cauchy property, and be able to deploy either when analysing the convergence or divergence of sequences;
- Apply knowledge of the theory and practice of power series expansions and Taylor polynomial approximations;
- Describe the role of Fourier series in representing periodic functions.
Generic skills
problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
collaborative skills: the ability to work in a team;
time management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 14 March 2025
Eligibility and requirements
Prerequisites
One of
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST10006 | Calculus 2 | No longer available | |
| MAST10021 | Calculus 2: Advanced | No longer available |
MAST10019 Calculus Extension Studies
Important: to qualify as prerequisites for this subject, MAST10006 and MAST10019 must be completed with a result of at least 75.
AND
One of
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST10007 | Linear Algebra | No longer available | |
| MAST10008 | Accelerated Mathematics 1 | No longer available | |
| MAST10022 | Linear Algebra: Advanced | No longer available |
MAST10018 Linear Algebra Extension Studies
Important: to qualify as prerequisites for this subject, MAST10007 and MAST10018 must be completed with a result of at least 75.
Corequisites
None
Non-allowed subjects
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST10009 | Accelerated Mathematics 2 | No longer available | |
| MAST20026 | Real Analysis | No longer available |
Inherent requirements (core participation requirements)
The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.
Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home
Last updated: 14 March 2025
Assessment
| Description | Timing | Percentage |
|---|---|---|
Five equally weighted written assignments due at regular intervals
| During the teaching period | 30% |
A written examination
| During the examination period | 70% |
Last updated: 14 March 2025
Dates & times
- Semester 1
Principal coordinator Jesse Gell-Redman Mode of delivery On Campus (Parkville) Contact hours 3 x one hour lectures per week; 2 x one hour practice classes per week. Total time commitment 170 hours Teaching period 26 February 2024 to 26 May 2024 Last self-enrol date 8 March 2024 Census date 3 April 2024 Last date to withdraw without fail 3 May 2024 Assessment period ends 21 June 2024 Semester 1 contact information
What do these dates mean
Visit this webpage to find out about these key dates, including how they impact on:
- Your tuition fees, academic transcript and statements.
- And for Commonwealth Supported students, your:
- Student Learning Entitlement. This applies to all students enrolled in a Commonwealth Supported Place (CSP).
Subjects withdrawn after the census date (including up to the ‘last day to withdraw without fail’) count toward the Student Learning Entitlement.
Last updated: 14 March 2025
Further information
- Texts
- Breadth options
- Available through the Community Access Program
About the Community Access Program (CAP)
This subject is available through the Community Access Program (also called Single Subject Studies) which allows you to enrol in single subjects offered by the University of Melbourne, without the commitment required to complete a whole degree.
Please note Single Subject Studies via Community Access Program is not available to student visa holders or applicants
Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.
- Available to Study Abroad and/or Study Exchange Students
Last updated: 14 March 2025