Further topics in analysis (MATH 11521)


In a nutshell

Cauchy

Starting: Monday, 27th January 2014
Times: Monday 2-3pm, Tuesday 1-2pm
Venue: Chemistry Building, Lecture Theatre 1
Level: 1st year undergraduate (10 credit points)
Prerequisites: Analysis I
Department information: Further Topics in Analysis (MATH 11521)
Office hours: Monday, 5-6pm (Howard House, Office 2.06)
 


Synopsis |  Syllabus |  Notes | Homework | Assessment | Books | Further links


Synopsis

This course builds on what you have learnt in Analysis I, and introduces you to the important notion of a Cauchy sequence as well as the concepts of uniform continuity and uniform convergence. In order to place these concepts on solid foundations, we will first spend a bit of time studying various notions of cardinality for finite and infinite sets. The last third of the course is dedicated to developing a rigorous theory of Riemann integration.

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Syllabus

The syllabus below is a rough indication only. The topics taught on any given day may change at short (or with no) notice.

Date ContentHomework
27/01/2014 relations and equivalence relationsSheet 1, Questions 1, 2, 3
28/01/2014 functions" "
03/02/2014 finite and countable setsSheet 2, Questions 1, 3, 4
04/02/2014 properties of countable sets" "
10/02/2014 uncountable sets and continuaSheet 3, Questions 1, 4, 5
11/02/2014 cardinality, Cantor-Schröder-Bernstein theorem" "
17/02/2014 hierarchy of cardinalitiesSheet 4, Questions 2, 3, 4
18/02/2014 subsequences and accumulation points" "
24/02/2014 Bolzano-Weierstrass theoremSheet 5, Questions 1, 3, 4
25/02/2014 limit superior and limit inferior" "
03/03/2014 characterisation of limit superior and limit inferiorSheet 6, Questions 4, 5, 6
04/03/2014 Cauchy sequences" "
10/03/2014 uniform continuitySheet 7, Questions 1, 2, 4
11/03/2014 uniform convergence" "
17/03/2014 Weierstrass's theorem on uniform convergenceSheet 8, Questions 1, 3, 5
18/03/2014 revision/examples" "
24/03/2014 definition of the Riemann integralSheet 9, Questions 2, 3, 4
25/03/2014 properties of the Riemann integral" "
31/03/2014 classes of integrable functionsSheet 10, Questions 2, 3, 4
01/04/2014 inequalities and the mean-value property" "
Easter Break
28/04/2014 fundamental theorem of calculusSheet 11, Questions 1, 2, 3
29/04/2014 revision/examples I:
uniform continuity and uniform convergence, Riemann integration
Supplementary exercises
05/05/2014 No lecture! No office hours! Bank Holiday
06/05/2014 revision/examples II:
countability, cardinality and accumulation points, liminf and limsup, Cauchy sequences
Supplementary exercises

Special office hours
I expect to hold a couple of special office hours for revision purposes. Please come prepared with questions. Here is a provisional timetable (HH stands for Howard House).

09/05/2014 Friday, 3-5pm2nd floor seminar room, HH
15/05/2014 Thursday, 11am-1pm2nd floor seminar room, HH
29/05/2014 Thursday, 11am-1pm2nd floor seminar room, HH

Exam information
Your exam takes place on

04/06/2014 Wednesday morning, 9:30-11:00amPithay Building, BS1 2NB NOTE CHANGE OF VENUE!

For further information please see the assessment section of this site.

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Notes

The lecture notes for the course can be downloaded in pdf format via the following link.

Further Topics in Analysis (Notes 2013-2014)

Please notify the lecturer of any typos, errors or omissions by email or in person.

Added 04/02/2014: The end of Section 3 has been slightly updated.
Added 28/02/2014: Typo fixed on page 28.

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Homework

You are required to attend all tutorials for this course. Written solutions to the set homework problems are to be handed in to your subject tutor in advance of the tutorial each week.

Problem sheets
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9
Week 10
Week 11

The above links will be activated as the course progresses. Solution sets will be posted below two weeks after the material was presented in lectures.

Solutions
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9
Week 10
Week 11

Midterm revision sheet
These short questions cover roughly the first two thirds of the course. They should help you get a sense for which topics you are already comfortable with and which topics will need your attention in the coming weeks.

Questions
Solutions

Supplementary exercises
Once you have gone over all the questions on the examples sheets (including those not originally set as homework), you may wish you try some of the supplementary exercises below. In addition, you will want to look at past exam papers, which are available on Blackboard (but see remarks under Assessment below).

Additional exercises
Solutions

Sample exam question
The following would make a suitable question for Section B of the exam, and the solution provided reflects the level of detail we expect to see for full marks.

Sample exam question
Sample solution

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Assessment

The course will be assessed by a 1.5h written exam in May/June, consisting of two sections. Section A contains 5 short questions carrying 40% of the marks, and Section B contains 2 longer questions accounting for the remaining 60% of marks. You will be expected to attempt all questions.

Your exam takes place at Bristol City Football Grounds on Wednesday, 4th June 2014, from 9:30am to 11:00am. Please make sure that you know how to get there!

Past examination papers are available for download on Blackboard. Note that the format of the exam has changed since last year in that all questions in Section B are now compulsory (but there are fewer of them). A sample exam question with a complete solution is posted at the bottom of the homework section above.

Detailed rules and regulations pertaining to exams may be found here.

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Books

The following texts may be useful.

•   C.W. Clark. Elementary mathematical analysis, Wadsworth Publishers of Canada, 1982.
•   E. Hairer and 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 and D. Tall. The foundations of mathematics, Oxford University Press, 1977.

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Further links

Fields medalist Tim Gowers has 'A brisk tutorial on countability' which you may find useful, as well as an excellent essay entitled 'Why easy analysis problems are easy'.

You should also consult his pages for a detailed treatment of some of those set-theoretic issues that are swept under the carpet in this course (and indeed almost any other introductory course in analysis). Of particular interest to those who like to worry about these things in their spare time may be 'What is wrong with thinking of real numbers as infinite decimals?', 'What is naive about naive set theory?' and 'Paradoxes concerning definability'. Finally, more information about the axiom of choice, which we implicitly assume in a number of places throughout the course, together with a discussion of why it is subtle, can be found on Eric Schechter's page or on wikipedia.

Vicky Neale has two excellent interactive exercises on her teaching pages. The first one is a multiple choice quiz that tests your understanding of the basic definitions and concepts of analysis. The second is a so-called proof-sorter exercise, which asks you to 'sort' two different proofs of the Bolzano-Weierstrass theorem (one of which I gave in lectures, the other one of which is in your notes, see also this summary).

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This page was last updated 8th May 2014.