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Further Topics in Analysis (MATH 11521)

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Administrative Information

  1. Unit number and title: MATH 11521 Further Topics in Analysis
  2. Level: C/4 (Honours)
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 1999/2000
  6. Unit Organiser: Jens Marklof
  7. Lecturer: Prof Jens Marklof
  8. Teaching block: 2
  9. Prerequisites: A at A-level Mathematics or equivalent. Co-requisite: Analysis 1.

Unit 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.

General Description of the Unit

The unit starts by extending some of the basic ideas concerning sets and functions which will have been introduced by this stage in Analysis 1. Then some further topics in analysis are covered, including additional material on sequences and series, and the theory of the Riemann integral.

Relation to Other Units

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

Teaching Methods

The course will be based on lectures and (for first year students) small group tutorials. Homework exercises will be marked by tutors or the lecturer and model solutions will be provided.

Learning Objectives

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.

Assessment Methods

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.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

The assessment mark is calculated as described in the Assessment section above. Details of the university's common criteria for the award of credit points are set out in the Regulations and Code of Practice for Taught Programmes at http://www.bristol.ac.uk/esu/assessment/codeonline.html

Note that for this unit:

  • first year students are expected to attend all the relevant tutorials,
  • all students are expected to hand in attempts to the weekly exercises set.

Transferable Skills

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

Texts

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 

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]