## Further topics in analysis (MATH 11521)## In a nutshell
## Synopsis | Syllabus | Notes | Homework | Assessment | Books | Further links## SynopsisThis 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. ## SyllabusThe syllabus below is a rough indication only. The topics taught on any given day may change at short (or with no) notice.
For further information please see the ## NotesThe lecture notes for the course can be downloaded in pdf format via the following link.
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.
## HomeworkYou are required to attend
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.
## AssessmentThe 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 Your exam takes place at Bristol City Football Grounds on 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 Detailed rules and regulations pertaining to exams may be found here. ## BooksThe following texts may be useful. • C.W. Clark. ## Further linksFields 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. | © 2003-2017 Julia Wolf |