This module is not currently running in 2024 to 2025.
This module will give an introduction to one of the main areas underpinning research in Analysis today: Functional Analysis, which has applications in many sciences, in particular in the modern theory of solutions of partial differential equations. As well as giving the main definitions and theorems in the area, the module will focus on applications, in particular to differential equations and in approximation theory. The following topics will be covered in the module: 1) Linear spaces: Normed and Banach spaces, Inner-product and Hilbert spaces, examples 2) Linear operators and functionals: bounded linear operators, functionals, dual spaces, reflexive spaces, adjoint operators, selfadjoint operators, examples 3) Fundamental theorems: Hahn-Banach, Uniform boundedness principle, Open mapping & Closed graph theorem, Baire Category theorem 4) Fixed point theorems and applications to differential and integral equations 5) Applications in approximation theory: best approximation in Hilbert space, approximation of continuous functions by polynomials. Possible additional topic: Spectral theory of bounded linear operators, weak and weak* topologies, algebras of bounded linear operators.
42-48 lectures and example classes
80% Examination, 20% Coursework
1) Introductory Functional Analysis with Applications, Erwin Kreyszig, John Wiley, 1978.
2) Principles of Mathematical Analysis. Walter Rudin, International Series in Pure and Applied Mathematics, McGraw-Hill, 1976 3rd edition.
3) Beginning Functional Analysis, Karen Saxe, Springer, 2002.
4) Introduction to Functional Analysis, Angus E.Taylor, David C.Lay, John Wiley, 1980 2nd edition.
See the library reading list for this module (Canterbury)
On successful completion of this module, students will:
a) be able to work with fundamental concepts in functional analysis, such as linear operators and functionals;
b) have a grasp of formal definitions and rigorous proofs in analysis;
c) have gained an appreciation of a wider context in which previously encountered concepts from analysis, such as convergence and continuity, can be used;
d) be able to apply abstract ideas to concrete problems in analysis;
e) appreciate differences between analysis in infinite and finite dimensional spaces;
f) be aware of applications of basic techniques and theorems of functional analysis in other areas of mathematics, e.g., approximation theory, and the theory of ordinary differential equations
University of Kent makes every effort to ensure that module information is accurate for the relevant academic session and to provide educational services as described. However, courses, services and other matters may be subject to change. Please read our full disclaimer.