Courses / Modules / MATH3076 Hilbert Spaces

Hilbert Spaces

When you'll study it
Semester 1
CATS points
ECTS points
Level 6
Module lead
Shintaro Nishikawa
Academic year

Module overview

This module is an introduction to the functional analysis of Hilbert spaces. The subject of functional analysis builds on the linear algebra studied in the first year and the analysis studied in the second year.

Initially pivotal in Fourier theory and differential equations, Hilbert spaces have evolved into the cornerstone of modern Quantum Mechanics and Quantum Information Theory.

The module begins by refreshing our understanding of linear algebra, focusing on key concepts such as orthonormal bases, self-adjoint operators, unitaries, and more. We will lay the groundwork for functional analysis on Hilbert spaces, delving into operator norms, spectrum, and the spectral theorem. Additionally, we will explore the tensor product of Hilbert spaces, and catch a glimpse of how these concepts tie into quantum information theory.

Building upon our finite-dimensional understanding, we will venture into infinite-dimensional Hilbert spaces. Here, we will introduce the concept of completeness for inner product spaces, and carefully examine how our foundational concepts extend to this setting. New concepts such as compact operators, unilateral shift operators, and continuous spectrum will be explored, along with the intriguing realm of unbounded operators. Lastly, we will touch upon Heisenberg uncertainty principle in Quantum Mechanics, connecting theory to real-world applications.

Linked modules

Pre-requisites: MATH1049 AND MATH2039