Handbook 1996 : Faculty of Science (Volume 4 page 209)
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618-201 Real Analysis (Advanced)

Credit points: 12.0

Coordinator: Dr J J Koliha

Prerequisite: Mathematics 618-112 or 618-211, or by invitation (See Note 2 below)

Contact: 39 lectures (three a week).

Timetable: First semester

Objectives:

On completion of this subject, students should:

Comprehend:

• the structure and methods of proof;
• the concept of convergence of sequences and series; elementary topology of the real line;
• the fundamentals of continuity, differentiability and integration; interchange of limiting operations and elementary properties of metric spaces.

Have developed:

• skills in constructing rigorous and accurate arguments;
• skills in determining the convergence or otherwise of sequences and series;
• an understanding of integration and how it is an extension of antidifferentiation;
• a knowledge of when limiting operations can be interchanged.

Appreciate:

• the importance and satisfaction of rigorous arguments via proofs;
• the fundamental concepts of topology of the real line and how they extend to more abstract settings;
• the inter-relationships of various branches of real analysis; applications of real analysis;
• the deeper and more abstract aspects of real analysis and how they can sometimes defy intuition.

Content:

Sequences: standard sequences, least upper and greatest lower bounds, Bolzano-Weierstrass theorem, upper and lower limits, Cauchy convergence. Elementary topology: open and closed sets, nested intervals, Heine - Borel theorem. Series: standard series, ratio and n-th root tests, absolute and conditional convergence, re-arrangements, power series. Continuity: sequential continuity, differentiability, uniform continuity, approximation of continuous function by step functions, introduction to Riemann integration. Metric spaces: examples of metric spaces, convergence. Uniform convergence: term-by-term operations on sequences and series, application to power series.

Assessment:

Up to 26 pages of written assignments and up to three hours of end-of-semester written examination.

Notes:

1. Credit cannot be obtained for both 618-201 and 618-300 (1995 Handbook).
2. Students with a high level of achievement in 618-102 (1995 Handbook) or 618-122 may approach the Head of the Department of Mathematics to seek permission to enrol in this subject.

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Handbook 1996 : Faculty of Science (Volume 4 page 209)

Status:          Official 1996
Date created:    Oct  9 1995
Last modified:   Oct  9 1995
Authorised by:   Academic Registrar
Email enquiries: Course_Information@registrar.unimelb.edu.au

Copyright © University of Melbourne 1995,1996.