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MAU34212 Functional Analysis

Credit weighting (ECTS)
5 credits
Semester/term taught
Semester 2
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof Florian Naef
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
MAU22200 Advanced analysis is desirable
Assessment Detail
This module will be examined in a 2 hour examination. Continuous assessment will contribute 20% to the final grade for the module at the annual examination session. Reassessments, if required, will consist of 100% exam.