Hilbert Basis

A Hilbert basis for the vector space of square summable sequences (a_n)=a_1, a_2, ... is given by the standard basis e_i, where e_i=delta_(in), with delta_(in) the Kronecker delta. Then

with sum|a_i|^2<infty. Although strictly speaking, the e_i are not a vector basis because there exist elements which are not a finite linear combination, they are given the special term "Hilbert basis."

In general, a Hilbert space V has a Hilbert basis e_i if the e_i are an orthonormal basis and every element v in V can be written

for some a_i with sum|a_i|^2<infty.

See also

Fourier Series, Hilbert Space, L2-Space, Orthonormal Set, Vector Basis

This entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Hilbert Basis." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HilbertBasis.html

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