# MA3G8 Functional Analysis 2

**Lecturer:** James Robinson

**Term(s):** Term 2

**Status for Mathematics students:** List A

**Commitment:** 30 lectures

**Assessment:** 3 hour examination (100%)

**Prerequisites:** MA3G7 Functional Analysis I, MA359 Measure Theory would be useful but is not required

**Leads To: **MA4A2 Advanced PDEs, MA433 Fourier Analysis, MA4G6 Calculus of Variations, MA4A2 Advanced PDEs

and MA4J0 Advanced Real Analysis.

**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.

**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.