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. The word 'functional' goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to Volterra.

Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism 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 (ℵ0) 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.

Related Topics:
Isomorphism - Cardinality - Linear algebra - Aleph-null

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