Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. It's essentially just an infinitely dimensional vector space. ﻿A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.
Hilbert transform
The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between "L"(R) and the dual space "L"(R), where "p" and "q" are Hölder conjugates and 1 Banach space of "real"-valued functions in "L"(R), the Hilbert transform defines a linear complex structure on this Banach space. In particular, when "p"=2, the Hilbert transform gives the Hilbert space of real-valued functions in "L"(R) the structure of a "complex" Hilbert space.