Hilbert Space

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

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).

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.