Content: This is essentially a module about infinite-dimensional Hilbert spaces, which arise naturally in many areas of applied mathematics. The ideas presented here allow for a rigorous understanding of Fourier series and more generally the theory of Sturm-Liouville boundary value problems. They also form the cornerstone of the modern theory of partial differential equations.
Hilbert spaces retain many of the familiar properties of finite-dimensional Euclidean spaces ( ) - in particular the inner product and the derived notions of length and distance - while requiring an infinite number of basis elements. The fact that the spaces are infinite-dimensional introduces new possibilities, and much of the theory is devoted to reasserting control over these under suitable conditions.
The module falls, roughly, into three parts. In the first we will introduce Hilbert spaces via a number of canonical examples, and investigate the geometric parallels with Euclidean spaces (inner product, expansion in terms of basis elements, etc.). We will then consider various different notions of convergence in a Hilbert space, which although equivalent in finite-dimensional spaces differ in this context. Finally we consider properties of linear operators between Hilbert spaces (corresponding to the theory of matrices between finite-dimensional spaces), in particular recovering for a special class of such operators (compact self-adjoint operators) very similar results to those available in the finite-dimensional setting.
Throughout the abstract theory will be motivated and illustrated by more concrete examples.
Books: An obvious recommendation would be:
JC Robinson, An Introduction to Functional Analysis, Cambridge University Press, 2020.
The module will follow parts of this book quite closely. However, if you want a different take on the material,
the book BP Rynne & MA Youngson, Linear Functional Analysis, Springer-Verlag, London, 2000 is very good.