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 .

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

**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.

© 1996-9

1999-05-25