# Course Name

Hilbert Spaces

Session: VDNS3121

18 Credit Points

# Prerequisites & Language Level

Taught In English

• There is no language prerequisite for courses at this language level.

# Overview

An introduction to Hilbert spaces and linear operators on Hilbert spaces, grounded in applications to Fourier analysis, spectral theory and operator theory.

MATH 301 extends the techniques of linear algebra and real analysis to study problems of an intrinsically infinite-dimensional nature. A Hilbert space is a vector space with an inner product that allows length and angles to be measured; the space is required to be complete (in the sense that Cauchy sequences have limits) so that the techniques of analysis can be applied. Hilbert spaces arise frequently in mathematics, physics, and engineering, often as infinite-dimensional function spaces. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (with applications to signal processing and heat transfer) and many areas of pure mathematics including topology, operator algebra and even number theory.

The course will introduce students to the basic techniques of functional analysis in the context of Hilbert spaces and linear operators on Hilbert spaces. The course will be grounded in applications to Fourier analysis, spectral theory and operator theory, will reinforce the students' understanding of linear algebra and real analysis, and will give them training in modern mathematical reasoning and writing.

Course Structure
Main topics
• Inner-product spaces, the Cauchy Schwarz inequality and the norm
• Cauchy sequences and completeness, examples of Hilbert spaces
• Normed spaces and bounded linear operators
• Closed subspaces and orthogonal projections, convexity and least squares approximation
• Orthonormal bases and the reconstruction formula
• The Fourier basis and Fourier series
• Uniform convergence and the Fourier series of smooth functions
• Diagonalisation of compact self-adjoint operators

Learning Outcomes

• To understand the basic techniques of functional analysis in the context of Hilbert spaces and linear operators on Hilbert spaces
• To gain experience in modern mathematical reasoning and writing.

*Course content subject to change