## Hilbert Space

A Hilbert space is Vector Space with an Inner Product such that the Norm defined by

turns into a Complete Metric Space. If the Inner Product does not so define a Norm, it 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

A Hilbert space is always a Banach Space, but the converse need not hold.

See also Banach Space, L2-Norm, L2-Space, Liouville Space, Parallelogram Law, Vector Space

References

Sansone, G. Elementary Notions of Hilbert Space.'' §1.3 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 5-10, 1991.

Stone, M. H. Linear Transformations in Hilbert Space and Their Applications Analysis. Providence, RI: Amer. Math. Soc., 1932.