# Module MA3421: Functional Analysis I

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2014-15
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof Richard Timoney
Learning Outcomes
On successful completion of this module, students will be able to:
• Give the appropriate definitions, theorems and proofs concerning the syllabus topics, including topics in general topology, elementary theory of Banach spaces and of linear operators;
• Solve problems requiring manipulation or application of one or more of the concepts and results studied;
• Formulate mathematical arguments in appropriately precise terms for the subject matter;
• Apply their knowledge in mathematical domains where functional analytic techniques are relevant;
Module Content
• General Topology: 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.
• Normed and Banach spaces: 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 ($1 \leq p \leq \infty$). Use of 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). $\ell^p$ increases with $p$ while $L^p([0,1])$ decreases and the inclusion maps as examples of operators.
• Baire category theorem and some of its consequences (open mapping). An application to Fourier series.

For further information refer to the module web pages.

Module Prerequisite
MA2223-Metric Spaces and MA2224-Lebesque Integral are desirable
Assessment Detail
This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the annual examination session.