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 proof, contrapositive, contradiction. The irrationality of root 2 and e. Natural numbers and induction: the binomial theorem; Functions: one-to-one, onto, bijections. Countability; Completeness of the real numbers; Topology of the reals: open and closed intervals, structure theorem of open sets, compact sets; Continuous functions; definitions and properties, intermediate value theorem, global extrema on compact sets.
Metric Spaces: introduction to metric spaces, with emphasis on function spaces and their properties: the space of continuous functions on an interval, uniform convergence; Arzela-Ascoli theorem; Stone-Weierstrass theorem.
Lebesgue integration: definition and properties, Fatou's lemma, monotone convergence theorem, dominated convergence.

Textbook(s)

Kolmogorov, A.N and Fomin, S.V 1975, Introductory Real Analysis, Dover Publications Inc

Prerequisite(s)

Common to all relevant programs
Subject Area & Catalogue Number	Course Name
MATH 1055	Calculus 2

Corequisite(s)

Nil

Teaching method

Component	Duration
INTERNAL, MAWSON LAKES
Lecture	3 hours x 13 weeks
Tutorial	1 hour x 12 weeks

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

Assessment

Assignment 1, Assignment 2, Examination

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
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* 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.