It is a mathematical concept.It is named after David Hilbert. It extendts the method of vector algebra from 2 dimensional and three dimensional plane to infinite dimensional plane.
http://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Hilbert.jpg/180px-Hilbert.jpg

A Hilbert space is a vector space with an inner product such that the norm defined by

turns into a complete metric space. If the metric defined by the norm is not complete, then is instead known as an inner product space.

Examples of finite-dimensional Hilbert spaces include

1. The real numbers with the vector dot product of and .

2. The complex numbers with the vector dot product of and the complex conjugate of .

An example of an infinite-dimensional Hilbert space is , the set of all functions such that the integral of over the whole real line is finite. In this case, the inner product is

From http://mathworld.wolfram.com/HilbertSpace.html
"A Hilbert space is always a Banach space, but the converse need not hold.

A (small) joke told in the hallways of MIT ran, "Do you know Hilbert? No? Then what are you doing in his space?" (S. A. Vaughn, pers. comm., Jul. 31, 2005).

SEE ALSO: Banach Space, Complete Set of Functions, Hilbert Basis, Inner Product Space, L2-Norm, L2-Space, Liouville Space, Parallelogram Law, Rigged Hilbert Space, Vector Space"
Source(s):
wikipedia
http://mathworld.wolfram.com/HilbertSpace.html

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