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 uniform convergence and the Riemann integral and leads on to the abstract concept of a metric space. Finally we look at some applications to solving equations via the fixedpoint method, and at the rudiments of Fourier series.
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 practise 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 
The course has a main section and a skills section. The main section consists of 3 weekly lectures and a weekly workshop designed to augment and extend understanding the material covered in the lectures in a smallergroup setting. There will be frequent reading assignments and exercise assignments which students will be expected to have completed before the lecture.
The syllabus for the main part of the course is:
Uniform convergence of sequences and series of functions, power series.
The Riemann integral: step functions, the class of Riemannintegrable functions and their integrals. Basic properties. Integration of continuous functions via uniform continuity and the fundamental theorem of calculus. Integration and uniform convergence.
Metric spaces: definition and examples, especially R^n and C([a,b]) with various metrics. Convergence, Cauchy sequences; completeness, examples and nonexamples. The sup metric on C([a,b]) and uniform convergence. Equivalent metrics. Point set topology: open balls, open sets, closed sets, interior, closure, boundary, limit points. Continuity. Connectedness. Compactness: examples and nonexamples; relationships between compactness, closedness and boundedness, HeineBorel theorem in R^n.
Banach contraction mapping theorem and applications to algebraic equations, differential equations and integral equations.
Fourier series.
Skills: The content will be chosen appropriate to the learning outcomes. (10h)

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:
Fundamentals of Pure Mathematics (MATH08064)

Corequisites  
Prohibited Combinations  
Other requirements  Students must not have taken :
MATH10008 Pure & Applied Analysis or MATH10049 Metric Spaces 
Information for Visiting Students
Prerequisites  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, Partyear 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
On completion of this course, the student will be able to:
 The ability to state definitions and principal results of the course accurately, to present and explain standard proofs, and to use these together with the principal concepts, arguments and methods presented in the course in standard situations, and also to tackle problems analogous to or extending seen examples.
 The ability to find, explain and discuss examples that illustrate the theory in the course or which do or do not satisfy a definition or the conditions or conclusions of a theorem.
 The ability to use the theory and methods of the course to address and solve unseen problems in both concrete and abstract situations.
 The ability to combine purposeful private study with help from fellow students, tutors and lecturers and to use lecture notes and alternative sources to build a sound basis of understanding of the areas of mathematical analysis presented in the course.
 Demonstrate the ability to read and write mathematics using advanced notation accurately and appropriately, demonstrating the role of axioms, definitions, conjectures, theorems etc. in mathematical practice.

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)

Additional Information
Graduate Attributes and Skills 
Not entered 
Study Abroad 
Not Applicable. 
Keywords  HAna 
Contacts
Course organiser  Prof A Carbery
Tel: (0131 6)50 5993
Email: A.Carbery@ed.ac.uk 
Course secretary  Mr Thomas Robinson
Tel: (0131 6)50 4885
Email: Thomas.Robinson@ed.ac.uk 

