Students
MATH7901 – Analysis
2024 – Session 1, In person-scheduled-weekday, North Ryde
General Information
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| Unit convenor and teaching staff |
Unit convenor and teaching staff
Catherine Penington
Ji Li
|
|---|---|
| Credit points |
Credit points
10
|
| Prerequisites |
Prerequisites
Admission to MRes
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| Corequisites |
Corequisites
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| Co-badged status |
Co-badged status
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| 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.
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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
Assessment Tasks
| Name | Weighting | Hurdle | Due |
|---|---|---|---|
| Assignment 1 | 20% | No | week5 |
| Assignment 2 | 20% | No | week7 |
| Assignment 3 | 20% | No | week9 |
| Assignment 4 | 20% | No | week11 |
| Assignment 5 | 20% | No | week13 |
Assignment 1
Assessment Type 1: Problem set
Indicative Time on Task 2: 4 hours
Due: week5
Weighting: 20%
Set of questions (with short answers required) in (1) Abstract integration, (2) the theory of metric space, (3) elementary properties of measures and integration
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
Assignment 2
Assessment Type 1: Problem set
Indicative Time on Task 2: 4 hours
Due: week7
Weighting: 20%
Set of questions (with short answers required) in (1) the Riesz representation theorem, (2) properties of Borel functions, (3) continuity properties of measurable functions
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
Assignment 3
Assessment Type 1: Problem set
Indicative Time on Task 2: 4 hours
Due: week9
Weighting: 20%
Set of questions (with short answers required) in (1) approximation by continuous functions, (2) convex function and inequalities, (3) Lebesgue space and application.
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
Assignment 4
Assessment Type 1: Problem set
Indicative Time on Task 2: 4 hours
Due: week11
Weighting: 20%
Set of questions (with short answers required) in (1) inner product and linear functionals, (2) orthogonal sets, (3) Fourier series .
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.
- 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 5
Assessment Type 1: Problem set
Indicative Time on Task 2: 4 hours
Due: week13
Weighting: 20%
Set of questions (with short answers required) in (1) Fourier series of continuous functions, (2) Hahn-Banach theorem, (3) Poisson integrals.
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.
- 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
- the Writing Centre 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
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Unit information based on version 2024.01R of the Handbook