Functional analysis

Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. It has its historical roots in the study of transformations such as the Fourier transform and in the study of differential and integral equations.

In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example are the Hilbert spaces, where the norm arises from an inner product. These spaces are of utmost importance in the mathematical formulation of quantum mechanics.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C* algebras.

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomophism for every cardinality of the base. Since finite-dimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph Null dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph Null, and its morphisms. One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper subspace which is invariant. Many special cases have already been proven.

Banach spaces are much more complicated than Hilbert spaces. There is no clear definition of what would constitute a base, for example. Some examples of Banach spaces are "all Lebesgue measurable functions where the integral of the p'th power is finite". In Banach spaces, a large part of the study involves the dual subspace: the space of all continuous linear functionals. Unlike in linear algebra, the dual of the dual is not always isomorphic to the original space. However, there is always a natural monomorphism from a space into its dual's dual.