Functional analysis

Functional Analysis is the branch of Mathematics dealing with complete normed vector spaces over the complex numbers or real numbers. Such spaces are called Banach spaces. When the norm is the result of an inner product, the space is called a Hilbert space. Hilbert spaces have simply geometry: 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 unproven hypothesis in Functional Analysis is 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. 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 finite 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.

This is near-incomprehensible, could you at least have some pointers to explain a) what you're talking about in the first sentence, and b) assuming that most of us aren't going to understand explanations anyway, at least some glimmering of who might use this and why it's important? --Robert Merkel