# MA3G8 Functional Analysis II

**Lecturer:** Dr Andras Mathe

**Term(s):** Term 2

**Status for Mathematics students:** List A

**Commitment:** 30 lectures

**Assessment:** 100% 3 hour examination

**Formal registration prerequisites: **None

**Assumed knowledge: **

- Normed spaces
- Banach spaces
- Lebesgue spaces
- Hilbert spaces
- Dual spaces
- Linear operators

MA260 Norms, Metrics & Topologies or MA222 Metric Spaces:

- Normed spaces
- Metric spaces
- Continuity
- Topological spaces
- Compactness
- Completeness

**Useful background:**

MA260 Norms, Metrics & Topologies and MA222 Metric Spaces:

- Nowhere dense sets
- Baire category theorem

- Lebesgue measure
- Measurable functions
- Integral with respect to a measure

**Synergies: **The following modules go well together with Functional Analysis II:

**Leads To: **The following modules have this module listed as **assumed knowledge** or **useful background:**

- MA427 Ergodic Theory
- MA433 Fourier Analysis
- MA4A7 Quantum Mechanics: Basic Principles and Probabilistic Methods
- MA4A2 Advanced Partial Differential Equations
- MA4J0 Advanced Real Analysis
- MA4M2 Mathematics of Inverse Problems
- MA4L9 Variational Analysis and Evolution Equations

**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.