Undergraduate Course: Honours Analysis (MATH10068)

Course Outline
School	School of Mathematics	College	College of Science and Engineering
Credit level (Normal year taken)	SCQF Level 10 (Year 3 Undergraduate)	Availability	Available to all students
SCQF Credits	20	ECTS Credits	10
Summary	Core course for Honours Degrees involving Mathematics. This is a second course in real analysis and builds on ideas in the analysis portion of Fundamentals of Pure Mathematics. The course begins by introducing the abstract concept of a metric space to study analytic ideas such as continuity, but the majority of the course is spent on using these ideas to explore the concrete situation in n real dimensions. In the 'skills' section of this course we develop and start to use some of the fundamental tools of a professional mathematician that are often only glimpsed in lecture courses. Mathematicians formulate definitions (rather than just reading other people's), they make conjectures and then try and prove or disprove them. Mathematicians find their own examples to illustrate their own and other people's ideas, and they find new ways of developing the theory and new connections. We will explore and practice these activities in the context of material drawn from some of the lectures in the course and related subjects. We will practise explaining mathematics and also consider 'metacognitive skills': the ability that an experienced mathematician has to step back from a calculation or problem, to 'zoom out' and consider whether it is developing well or whether perhaps there is a flaw in the approach. A typical example is the habit stopping and asking whether a proof one is working on is actually using all the assumptions of the theorem.
Course description	A suggested syllabus for this course is: Weeks 1-4: The metric spaces structure of the real line. This is to include a review of Fundamentals of Pure Maths 2 (including sets and sequences), the definition of a metric space; limits, interior, compactness, and connectedness in a metric space, with emphasis on the situation in the real line; and previously omitted material on Cauchy sequences and uniform continuity in the real line. Weeks 5-6: Riemann integration. Weeks 7-8: Series of functions (including the completeness of C[a,b]). Week 9: Other metrics on C[a,b]. Weeks 10-11: Fourier series. Skills: The content will be chosen appropriate to the learning outcomes. (10h)

Entry Requirements (not applicable to Visiting Students)
Pre-requisites	Students MUST have passed: Fundamentals of Pure Mathematics (MATH08064)	Co-requisites
Prohibited Combinations		Other requirements	Students must not have taken : MATH10008 Pure & Applied Analysis or MATH10049 Metric Spaces

Information for Visiting Students
Pre-requisites	None
High Demand Course?	Yes

Course Delivery Information

Academic year 2015/16, Available to all students (SV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 200 ( Lecture Hours 35, Seminar/Tutorial Hours 10, Supervised Practical/Workshop/Studio Hours 10, Summative Assessment Hours 3, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 138 )
Assessment (Further Info)	Written Exam 80 %, Coursework 20 %, Practical Exam 0 %
Additional Information (Assessment)	Coursework 20%, Examination 80%
Feedback	Not entered
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S2 (April/May)	Honours Analysis (MATH10068)		3:00

Academic year 2015/16, Part-year visiting students only (VV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 200 ( Lecture Hours 35, Seminar/Tutorial Hours 10, Supervised Practical/Workshop/Studio Hours 10, Summative Assessment Hours 3, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 138 )
Assessment (Further Info)	Written Exam 80 %, Coursework 20 %, Practical Exam 0 %
Additional Information (Assessment)	Coursework 20%, Examination 80%
Feedback	Not entered
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S1 (December)	Honours Analysis (MATH10068)		3:00

Learning Outcomes
1. An ability to perform simple abstract arguments involving metric spaces. 2. An ability to demonstrate an understanding of notions such as openness, closedness, continuity, completeness, equivalence of metrics, compactness and path-connectedness as applied in the context of general and specific metric spaces. 3. Facility in working with concrete metric spaces based upon real ndimensional space and the discrete metric. In particular, able to work with open, closed, connected, and compact sets in real n-dimensional space and to work with continuous functions in real n-dimensional space using topological notions. 4. Understanding the definitions and concepts of convergence, Cauchy sequences, and completeness in R 5. Ability to test sequences and series of functions for uniform convergence, including the Weierstrass M test. 6. Familiarity with the idea that problems of analysis such as convergence of Fourier series can be couched in and resolved using the language of metric spaces. 7. Familiar with the supremum metric and with other metrics on the set of continuous functions and able to demonstrate an understanding of the inequivalence of these metrics. 8. Experience of mathematical activities such as formulating and exploring definitions and conjectures, identifying interesting questions and constructing illustrative examples. 9. Improved abilities in talking about and explaining mathematics 10. An awareness of and some development in a range of metacognitive skills in a mathematical context, for example considering the relation of a proof to the conditions of a theorem, identifying the key points in a proof or choosing to work on a simplified example of a problem first. 11. Deeper understanding of some representative material from lectures. 12. Experience of mathematical activities such as formulating and exploring definitions and conjectures, identifying interesting questions and constructing illustrative examples.

Reading List
Students are expected to have a personal copy of : Wade, W R, 'An Introduction to Analysis', 4th Edition, ISBN 9780136153702 (continuing students should already have a copy from Year 2)