Unit name | Functional Analysis 34 |
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Unit code | MATHM6202 |
Credit points | 20 |
Level of study | M/7 |
Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
Unit director | Dr. Bothner |
Open unit status | Not open |
Pre-requisites |
MATH20006 Metric Spaces |
Co-requisites |
None |
School/department | School of Mathematics |
Faculty | Faculty of Science |
Unit Aims
The unit aims to provide students with a firm grounding in the theory and techniques of functional analysis and to offer students ample opportunity to build on their problem-solving ability in this area.
Unit Description
This course sets out to explore some core notions in Functional Analysis. The focus of Functional Analysis is the study of infinite-dimensional vector spaces and the space of linear functions defined on an infinite-dimensional vector space. These spaces are endowed with some structures, e.g. norm, inner product, etc. Dealing with an infinite-dimensional space opens new possibilities, for example, Cauchy sequences may not converge. Banach spaces are spaces in which all Cauchy sequences converge. However, these spaces are in general very different from finite dimensional vector spaces. Hilbert spaces are an important subfamily of Banach spaces in which many of the familiar properties of finite-dimensional vector spaces hold.
Students will be introduced to the theory of Banach and Hilbert spaces. We reassert similar results to those studied in Linear Algebra in the finitedimensional setting. The highlight of the course will be an exposition of the four fundamental theorems in the Functional Analysis (Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, closed graph theorem). The unit may also include some discussion of the spectral theory of linear operators.
Functional analysis is helpful in the study of integral/differential equations and more general equations for operators in infinite dimensional spaces. It has found broad applicability in diverse areas of mathematics, physics, economics, and other sciences.
Relation to Other Units
This is a Level 7 version of the Level 6 unit Functional Analysis 3, and students may not take both units.
Learning objectives
By the end of the unit, students will
Transferable skills
Deductive thinking; problem-solving; mathematical exposition
The unit will be taught through a combination of
Recommended
Further