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)
Pre-requisites	Students MUST have passed: Honours Analysis (MATH10068) Students MUST have passed: Honours Algebra (MATH10069)	Co-requisites
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
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: 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, Part-year 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.