# Mathematics MA3421, Functional Analysis I

## Notes

Some (not all) parts of the course notes will be in the form of a handout or will be available here. All will be in PDF format and require a programme such as Adobe Acrobat Reader to read them.

- Chapter 1, Bases for topologies
- Review of metric spaces, definition of topological space, open/closed/boundary, continuity, limits of sequences, compactness, bases, second countability, separability, sub-bases, weak and product topologies, neighbourhood bases, first countability.
- Chapter 2, Banach spaces I
- Definitions and examples for Banach spaces and bounded linear operators. Examples include some `classical' Banach spaces $C(X)$, $\ell^p$, $c_0$, $L^p([0,1])$ and more general $L^p$ examples ($1 \leq p \leq \infty$). We use series in Banach spaces (convergent or absolutely convergent), basic concepts from Lebesgue integration. we show Hölder's and Minkowski's inequalities (vesions for sums and integrals). We show that $\ell^p$ increases with $p$ while $L^p([0,1])$ decreases and use the inclusions map as examples of operators.
- Chapter 3, Baire category and open mapping theorems
- We prove the Baire category theorem and use it to show the Open Mapping Theorem (for surjective linear operators between Banach spaces). We outline an application of the latter to Fourier series.