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 = δ_(i n), with δ_(i n) the Kronecker delta. Then (a_n) = sum a_i e_i, with sum( left bracketing bar a_i right bracketing bar )^2<∞. 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."

Fourier series | Hilbert space | L^2-space | orthonormal set | vector basis