Fundamentals of Real Analysis

Course aim

This course aims to introduce the student to fundamental concepts of analysis that underpin a large number of areas of mathematics, both pure and applied. By studying the material in this course the student will learn theoretical principles and techniques of analysis and learn to apply them so as to analyse mathematical problems.

Course content

Nature of proof in analysis: direct, contrapositive, and by contradiction.
The structure of real numbers: Completeness and arquemedian properties.
Irrationality of square root of 2.
Functions: one-to-one, onto, bijections.
Cardinality: Finite, and infinite sets, Countable and uncountable sets.
Topology of the reals: open and closed intervals, structure theorem of open sets, compact sets.
Metric Spaces and normed spaces: function spaces and their general properties.
Continuous functions; definitions and properties, intermediate value for continuous functions.
Differentiable functions: intermediate value theorem for differentiable functions, Taylor development for higher order differentiable functions, and applications to global extrema.
The space of continuous functions on an interval and on a compact set, uniform convergence; Arzela-Ascoli theorem; Stone-Weierstrass theorem.

Textbook(s)

Abbott, Stephen 2015, Understanding Analysis

Prerequisite(s)

Corequisite(s)

Nil

Teaching method

Note: These components may or may not be scheduled in every study period. Please refer to the timetable for further details.

Assessment

Examination, Problem solving exercise

Fees

EFTSL*: 0.125
Commonwealth Supported program (Band 2)
To determine the fee for this course as part of a Commonwealth Supported program, go to:
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Fee-paying program for domestic and international students
International students and students undertaking this course as part of a postgraduate fee paying program must refer to the relevant program home page to determine the cost for undertaking this course.

Non-award enrolment
Non-award tuition fees are set by the university. To determine the cost of this course, go to:
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Not all courses are available on all of the above bases, and students must check to ensure that they are permitted to enrol in a particular course.

* Equivalent Full Time Study Load. Please note: all EFTSL values are published and calculated at ten decimal places. Values are displayed to three decimal places for ease of interpretation.