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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Functional and Real Analysis (MATH11134)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 11 (Year 5 Undergraduate) AvailabilityAvailable to all students
SCQF Credits20 ECTS Credits10
SummaryThis course will cover the foundations of functional analysis in the context of normed linear spaces The Big Theorems (Hahn-Banach, Baire Category, Uniform Boundedness, Open Mapping and Closed Graph) will be presented and several applications will be analysed. The important notion of duality will be developed in Banach and Hilbert spaces and an introduction to spectral theory for compact operators will be given.

This course also introduces the essentials of modern real analysis which emerged from the work of Hardy and Littlewood in the 1930's and later from the work of Calderon and Zygmund in the 1950's. Many results and techniques from modern real analysis have become indispensable in many areas of analysis, including partial differential equations.
Course description - Review of linear spaces and their norms.
- The Hahn-Banach, Baire Category, Uniform Boundedness Principle, Open Mapping and Closed Graph theorems.
- Duality in Banach and Hilbert spaces.
- Spectral theory for compact operators on Hilbert spaces. Fredholm alternative
- Weak topologies, Banach-Alaoglu and the Arzela-Ascoli theorem.
- Covering lemmas, maximal functions and the Hilbert transform.
- The Fourier Transform, L1 and L2 theory.
- Weak type estimates and Interpolation
- Introduction to singular integrals and connections with partial differential equations.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Honours Analysis (MATH10068) AND Linear and Fourier Analysis (MATH10081) AND Essentials in Analysis and Probability (MATH10047)
Prohibited Combinations Students MUST NOT also be taking Functional Analysis (MATH11135) OR Real Analysis (MATH11136)
Other requirements None
Information for Visiting Students
Course Delivery Information
Not being delivered
Learning Outcomes
1. Facility with the main, big theorems of functional analysis.
2. Ability to use duality in various contexts and theoretical results from the course in concrete situations.
3. Capacity to work with families of applications appearing in the course, particularly specific calculations needed in the context of Baire Category.
4. Facility with the maximal functions and simple singular integrals.
5. Ability to use interpolation to reduce the study of certain linear and sublinear operators to their endpoint bounds.
6. Capacity to identify the essential features in methods and arguments introduced in the course and adapt them to other settings.
7. Be able to produce examples and counterexamples illustrating the mathematical concepts presented in the course.
8. Understand the statements and proofs of important theorems and be able to explain the key steps in proofs, sometimes with variation.
Reading List

1. Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. Universitext, Springer.

2. Elements of Functional Analysis, by Robert Zimmer, University of Chicago Lecture Series.

3. Singular Integrals and Differentiability Properties of Functions, by E.M. Stein, Princeton University Press.

4. Fourier Analysis, by J. Duoandikoetxea, Graduate Studies in Mathematics, Amer. Math. Soc.
Additional Information
Graduate Attributes and Skills Not entered
Course organiserDr Thomas Leinster
Tel: (0131 6)50 5057
Course secretaryMrs Alison Fairgrieve
Tel: (0131 6)50 5045
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