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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

 (a_n)=suma_ie_i,

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

 v=sum_(i=1)^inftya_ie_i

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:

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

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