Undergraduate Course: Linear Analysis (MATH10082)
Course Outline
School  School of Mathematics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 10 (Year 4 Undergraduate) 
Availability  Available to all students 
SCQF Credits  10 
ECTS Credits  5 
Summary  In this course, we will introduce students to techniques and tools in modern analysis which have important uses in a variety of areas of analysis, including the study of partial differential equations and fourier analysis.
We will achieve this in the context of linear analysis, introducing normed linear, inner product spaces and their completions, Banach and Hilbert spaces. The structure and geometry of these spaces will be studied as well as continuous linear operators acting on them. Many examples will be studied as well as connections to other fields. 
Course description 
 Inner product spaces and normed spaces.
 Completeness and completions of spaces with concrete realisations of standard examples. Lp spaces, Holder and Minkowski inequalities.
 Geometric and metric properties of Hilbert spaces, including orthonormal bases and generalised Fourier series.
 Bounded linear functionals, operators and duality,
 Test functions and distributions.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:
Honours Analysis (MATH10068) Students MUST have passed:
Honours Algebra (MATH10069)

Corequisites  
Prohibited Combinations  Students MUST NOT also be taking
Linear and Fourier Analysis (MATH10081)

Other requirements  Students might find it useful to have taken, or be taking, MATH10047 Essentials in Analysis and Probability.
Students wishing to take both MATH10082 Linear Analysis and MATH10051 Fourier Analysis in the same academic session should register for the 20 credit course MATH10081 Linear and Fourier Analysis. 
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:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )

Assessment (Further Info) 
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework 5%, Examination 95% 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S2 (April/May)  MATH10082 Linear Analysis  2:00  

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:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )

Assessment (Further Info) 
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework 5%, Examination 95% 
Feedback 
Not entered 
No Exam Information 
Learning Outcomes
On completion of this course, the student will be able to:
 Facility with the interplay between analysis, geometry and algebra in the setting of Banach and Hilbert spaces, both abstractly and in specific examples.
 Ability to use orthogonality arguments in a variety of theoretical and concrete situations.
 Capacity to work with the classes of normed linear spaces appearing in the course, particularly specific calculations around Hilbert spaces and operators acting on them.
 Be able to produce examples and counterexamples illustrating the mathematical concepts presented in the course.
 Understand the statements and proofs of important theorems and be able to explain the key steps in proofs, sometimes with variation.

Reading List
1. An Introduction of Hilbert Space, by N. Young, Cambridge Mathematical Textbooks.
2. Introduction to Hilbert Space, by S. Berberian, Oxford University Press. 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  LAn 
Contacts
Course organiser  Prof A Carbery
Tel: (0131 6)50 5993
Email: A.Carbery@ed.ac.uk 
Course secretary  Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk 

