Scottish Mathematical Sciences Training Centre

Theme: Analysis
Theme head: Dr Heiko Gimperlein

Module Overview: Functional Analysis

This module covers the following topics:

• Banach and Hilbert spaces
• Linear operators and linear functionals
• Fundamental theorems: Baire Category, Open Mapping, Uniform Boundedness Principle,...
• Weak and weak* topologies
• Operator topologies
• Krein-Milman Theorem (very useful convexity result)
• Compact operators and their spectra
• General spectral theory: Banach algebras, Spectral theorem for self-adjoint operators
• Commutative C*-algebras – Gelfand’s theory

Weekly Preparation

Students should read and study the lecture notes prior to the lecture time and come prepared to discuss and ask questions about topics covered in the notes.

Assessment

This module is assessed in two assignments with three questions per assignment. Due middle and end of module.

Prerequisites

• Undergraduate analysis: Sequences, series, pointwise and uniform convergence
• Metric space topology: At least in Rd, continuity of functions, open, closed and compact sets
• Countable sets
• Some of the examples from the module will draw upon knowledge from the first semester module 'Measure and Integration', so some familiarity with measure theory will be useful to get the most out of the module. However explicit use of measure theory will not be required in the assessments.