Short answers, pulled from the story.
Functional analysis is a branch of mathematical analysis centered on vector spaces endowed with a limit-related structure, such as an inner product, a norm, or a topology, together with the linear functions defined on those spaces that respect the structure. In modern texts it is seen as the study of vector spaces endowed with a topology, especially infinite-dimensional spaces.
The general concept of a functional was introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The word functional as a noun was first used in Hadamard's 1910 book on the calculus of variations.
The four major theorems sometimes called the four pillars of functional analysis are the Hahn-Banach theorem, the open mapping theorem, the closed graph theorem, and the uniform boundedness principle, also known as the Banach-Steinhaus theorem.
Banach spaces are complete normed vector spaces over the real or complex numbers, while a Hilbert space is a Banach space whose norm arises from an inner product. Hilbert spaces can be completely classified up to isomorphism by the cardinality of the orthonormal basis, but general Banach spaces cannot be classified so simply and many lack any notion analogous to an orthonormal basis.
The uniform boundedness principle, also known as the Banach-Steinhaus theorem, was first published in 1927 by Stefan Banach and Hugo Steinhaus. It was also proven independently by Hans Hahn.
The invariant subspace problem is an open problem in functional analysis that asks whether every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of the problem have already been proven.