Students
MATH7901 – Analysis
2020 – Session 1, Weekday attendance, North Ryde
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Students should consult iLearn for revised unit information.
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General Information
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| Unit convenor and teaching staff |
Unit convenor and teaching staff
Unit Convenor/Lecturer
Ji Li
12WW 710
please refer to iLearn
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|---|---|
| Credit points |
Credit points
10
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| 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
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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
Coronavirus (COVID-19) Update
Assessment details are no longer provided here as a result of changes due to the Coronavirus (COVID-19) pandemic.
Students should consult iLearn for revised unit information.
Find out more about the Coronavirus (COVID-19) and potential impacts on staff and students
General Assessment Information
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Delivery and Resources
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Main textbook: W. Rudin’s “Real and complex analysis”
Unit Schedule
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The first five chapters of W. Rudin’s book “Real and complex analysis”:
Chapter 1: Abstract Integration
Chapter 2: Positive Borel measures
Chapter 3: Lp spaces
Chapter 4: Elementary Hilbert space theory
Chapter 5: Examples of Banach space techniques
Each of Chapters 1, 2, 4 and 5 takes an average of 5 hours lecturing, and Chapter 3 takes 4 hours.
The last week is for any unforseen delay.
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