This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered include:

Hilbert spaces - bounded linear operators, compact operators, the spectral theorem for compact self-adjoint operators; Fourier transform, applications to partial differential equations and the central limit theorem.

Measure theory - abstract measure theory, integration,  Fubini-Tonelli theorem, Radon-Nikodym theorem, Hausdorff measure, fractals.

Banach spaces and linear operators - basic properties, Baire category theorem and its consequences (uniform boundedness principle, closed graph and open mapping theorems), Hahn-Banach theorem and dual spaces, sequential version of Banach-Alaoglu theorem, dual spaces of L^p spaces and spaces of continuous functions. Applications to Fourier series, fractals.

Note: This is an HPC. It emphasises mathematical rigour and proof and continues the development of modern analysis from an abstract viewpoint.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of functional analysis and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of functional analysis techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from functional analysis
4. Apply problem-solving using functional analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts

Assessment will be based on:

• Four assignments (40% total; LO 1-4)
• Essay paper (15%; LO 1-4)
• Attendance and participation in lectures and tutorials (5%; LO 1-4)
• Take home exam (40%; LO 1-4)

36 lectures, tutorials by arrangement

A mark of 60 or more in MATH3320.

with MATH3022.