MA4001: Functional Analysis

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA4001
External Subject Code G100
Number of Credits 20
Level L7
Language of Delivery English
Module Leader Professor Marco Marletta
Semester Autumn Semester
Academic Year 2015/6

Outline Description of Module

This module is the natural successor to MA3005 Functional and Fourier Analysis and extends the ideas from MA3005 in a number of directions. While the nature of much of the material is quite abstract, the first text in the recommended reading for this module makes it clear that the mathematics in this module is essential for much of modern applied mathematics, including approximation theory and the study of linear and nonlinear partial differential equations. The emphasis in this module is on showing that by developing powerful abstract tools, concrete results which would otherwise be exceptionally difficult and messy to prove can be established by elegant methods suitable for the `lazy' mathematician.

On completion of the module a student should be able to

The student should have some knowledge of each of the following topics:

  • point set topology;
  • linear functionals;
  • linear operators;
  • approximation theory.

How the module will be delivered

30 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 120 hours of private study, including preparation of worked solutions for problem classes.

Skills that will be practised and developed

Skills:

The student should be able to apply the theorems from the course to solve concrete problems involving differential and integral operators.

Transferable Skills

Achieve precision and clarity of exposition.

Appreciate the power of abstract approaches to the solution of concrete problems.

How the module will be assessed

Formative assessment is carried out by means of regular exercises. Feedback to students on their solutions and their progress towards learning outcomes is provided during problem classes.

The summative assessment is the written examination at the end of the module. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 100 Functional Analysis 3

Syllabus content

  • Point set topology.
    • Defintions, examples, Minkowski and Holder inequalities, topologies and metrics. Banach spaces, l^p spaces, Hilbert spaces, orthogonal complements.
  • Linear functionals.
    • Dual spaces, continuity conditions, Zorn's Lemma, Hahn Banach theorem, linear functionals on Hilbert spaces, examples of dual spaces, weak and weak-* topologies, Alaoglu's theorem.
  • Linear Operators.
    • Continuity, definition of B(V,V'), isometries, separable Hilbert spaces, adjoint operators for Banach and Hilbert spaces. Baire Category Theorem, Open Mapping Theorem, Uniform Boundedness Principle. Inverses to bounded linear operators. Some elementary spectral theory.
  • Approximation of Continuous Functions.
    • Weierstrass theorem, algebras, lattices, Stone-Weierstrass theorem.

Essential Reading and Resource List

Analysis for Applied Mathematics, Cheney, W., Springer, 2000.

Basic Methods of Linear Functional Analysis, Pryce, J.D., Hutchinson, 1973.

Real and Complex Analysis, Rudin, W., McGraw-Hill, 1986.

Background Reading and Resource List

Not applicable.


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