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.
In academic year 202223 and later, MATH10101 Metric spaces is recommended for this course. 
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,
Spectral Theorem for compact, selfadjoint operators on a Hilbert space.

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  
Other requirements  Students will find it useful to have taken, or be taking, MATH10047 Essentials in Analysis and Probability.
In academic year 202223 and later, MATH10101 Metric spaces is recommended for this course.

Information for Visiting Students
Prerequisites  Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.

High Demand Course? 
Yes 
Course Delivery Information

Academic year 2020/21, 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
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)  MATH10082 Linear Analysis  2:00  

Academic year 2020/21, 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 20%, Examination 80% 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S1 (December)   2:00  
Learning Outcomes
On completion of this course, the student will be able to:
 Demonstrate facility with the interplay between analysis, geometry and algebra in the setting of Banach and Hilbert spaces, both abstractly and in specific examples.
 Use orthogonality arguments in a variety of theoretical and concrete situations.
 Work with the classes of normed linear spaces appearing in the course, particularly specific calculations around Hilbert spaces and operators acting on them.
 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 Jim Wright
Tel: (0131 6)50 8570
Email: J.R.Wright@ed.ac.uk 
Course secretary  Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk 

