Hilbert Space

A Hilbert space is Vector Space $H$ with an Inner Product $\left\langle{f,g}\right\rangle{}$ such that the Norm defined by

$$\vert f\vert = \sqrt{\left\langle{f,f}\right\rangle{}}$$

turns $H$ 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 $\Bbb{R}^n$ with $\left\langle{v,u}\right\rangle{}$ the vector Dot Product of $v$ and $u$.

2. The Complex Numbers $\Bbb{C}^n$ with $\left\langle{v,u}\right\rangle{}$ the vector Dot Product of $v$ and the Complex Conjugate of $u$.

An example of an Infinite-dimensional Hilbert space is $L^2$, the Set of all Functions $f:\Bbb{R}\to\Bbb{R}$ such that the Integral of $f^2$ over the whole Real Line is Finite. In this case, the Inner Product is

$$\left\langle{f,g}\right\rangle{} = \int f(x)g(x)\,dx.$$

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.