Content: 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.
In the final part of the course we will study the theory of distributions ("generalised functions") which allows one to make rigorous sense of the Dirac delta "function" and is a fundamental part of the modern theory of partial differential equations.
Books: Useful books to use as an accompanying reference to your lecture notes are:
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.