Introductory description

Problems posed in infinite-dimensional space arise very naturally throughout mathematics, both pure and applied. In this module we will concentrate on the fundamental results in the theory of infinite-dimensional Banach spaces (complete normed linear spaces) and linear transformations between such spaces.

We will prove some of the main theorems about such linear spaces and their dual spaces (the space of all bounded linear functionals) - e.g.
the Hahn-Banach Theorem and the Principle of Uniform Boundedness - and show that even though the unit ball is not compact in an infinite-dimensional space, the notion of weak convergence provides a way to overcome this.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Dual of sequence spaces; Hahn-Banach and separation theorems; Krein-Milman theorem; reflexivity; uniform boundedness principle; open mapping and closed graph theorems; weak and weak star convergence; Banach-Alaoglu theorem; Stone-Weierstrass theorem.

Learning outcomes

By the end of the module, students should be able to:

• By the end of the module students should be able to demonstrate deep knowledge and understanding of core areas of functional analysis and their applications to other fields of mathematics.

Indicative reading list

E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1989.
W. Rudin, Functional Analysis, McGraw-Hill, 1973.
G. B. Folland, Real Analysis, Wiley, 1999.
E.H. Lieb and M. Loss, Analysis, 2nd ed. American Mathematical Society, 2001.

Subject specific skills

Students will develop a broader understanding of mathematical analysis by studying the interactions of linear algebra, real analysis, metric spaces and measure theory in this module.

Transferable skills

This modules will further enhance students' analytical, problem-solving and reasoning skills.