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Hilbert Space


A Hilbert space is a vector space H with an inner product <f,g> such that the norm defined by

 |f|=sqrt(<f,f>)

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

Examples of finite-dimensional Hilbert spaces include

1. The real numbers R^n with <v,u> the vector dot product of v and u.

2. The complex numbers C^n with <v,u> 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:R->R such that the integral of f^2 over the whole real line is finite. In this case, the inner product is

 <f,g>=int_(-infty)^inftyf(x)g(x)dx.

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, Vector Space Explore this topic in the MathWorld classroom

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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.

Referenced on Wolfram|Alpha

Hilbert Space

Cite this as:

Weisstein, Eric W. "Hilbert Space." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HilbertSpace.html

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