Functional Analysis

Content

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Description

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
• Convexity and the Krein-Milman Theorem
• The Banach-Stone Theorem
• The Markov-Kakutani Theorem
• 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 take-home assignments with three questions per assignment:

Assignment 1
Released: Week 6
Due: Two weeks after release date

Assignment 2
Released: Week 10, last lecture
Due: Two weeks after release date

Assignments are weighted equally and each counts for 50% of the final grade.

Resources

• Chapters 2-5 in Rudin’s Real and Complex Analysis
• Chapters 1-4 in Stein & Shakarchi’s Real Analysis and Functional Analysis  books (books III and IV in their Princeton Lecture Series in Analysis)
• Chapters 1-6 in Arbogast & Bona’s ‘M383C notes’ Methods of Applied Mathematics. See also Debray’s write-up of semester 1
• Chapters 1, 3 & 4 in Klainerman’s PDEs notes