Unit information: Further Topics in Analysis in 2013/14

Unit name	Further Topics in Analysis
Unit code	MATH11521
Credit points	10
Level of study	C/4
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Professor. van den Berg
Open unit status	Not open
Pre-requisites	Normally and A at A level Maths or equivalent
Co-requisites	MATH11006 Analysis 1
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

The unit starts by extending some of the basic ideas concerning sets and functions which will have been introduced by this stage in the Analysis element of Core Mathematics, and goes on to such new topics as Russell's Paradox, equivalence relations, and countability and cardinality. Then some further topics in analysis are covered, including additional material on dequences, series, and continuous fractions.

Aims:

This unit aims to develop students' ability to think and express themselves in a clear logical fashion, and to develop some of the material on set theory which will have been introduced in Analysis 1, and some additional topics in analysis.

Syllabus

Further Set Theory: finite and countable sets, equivalence relations, cardinality, examples of uncountable sets, a set and its power set have different cardinalities, Russell's Paradox. [6 lectures]

Further Analysis: construction of the real numbers, subsequences, limit superior and limit inferior, Cauchy sequences, uniformly continuous functions, sequences and series of functions, the Riemann integral. [16 lectures]

Relation to Other Units

This unit complements the Analysis 1 material, and is a prerequisite for analysis units in later years.

Intended Learning Outcomes

After taking this unit students should:

• be able to understand and write clear mathematical statements and proofs;

• understand and be able to apply the basic concepts and results presented throughout the unit;

• be able to solve standard types of problems concerning sets, sequences, and series.

Teaching Information

The ability to express intuitive ideas in a precise mathematical fashion and to produce clear logical arguments.

Assessment Information

The final mark for Further Topics in Analysis is calculated from one 1½ -hour written examination in May/June. This examination paper is in two sections.

• Section A contains 5 short questions, ALL of which should be attempted. Section A contributes 40% of the mark for this paper.

• Section B has 3 longer questions; you should attempt TWO. If you attempt more than two, your best two answers in Section B will be used for assessment. Section B contributes 60% to the mark for this paper.

Calculators may NOT be used.

Reading and References

The following are useful but not essential:

• C.W. Clark, Elementary mathematical analysis, Wadsworth Publishers of Canada, 1982

• E. Hairer & G. Wanner, Analysis by its history, Springer-Verlag, 1996

• J. M. Howie, Real analysis, Springer-Verlag, 2001

• S. G. Krantz, Real analysis and foundations, Chapman & Hall/CRC Press, 1991

• I. Stewart & D. Tall, The foundations of mathematics, Oxford University Press, 1977