A Hilbert basis for the vector space of square summable sequences , , ... is given by the standard basis , where , with the Kronecker delta. Then
with . Although strictly speaking, the 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 has a Hilbert basis if the are an orthonormal basis and every element can be written
for some with .