A Hilbert basis of a Hilbert space 
is an orthonormal basis 
for which every element 
has an expansion converging in
the norm of 
,
 |
with 
.
For each 
,
at most countably many coefficients 
are nonzero. If 
is a separable space, the
orthonormal basis can be indexed by a finite
set or countable set. A nonseparable Hilbert
space requires an uncountable set of indices 
.
For example, a Hilbert basis for the vector space of square summable sequences 
, 
, ... is given by the standard
basis 
,
where the 
th
entry of 
is 
, the Kronecker
delta. Then
 |
with 
.
The 
are not a vector basis, because some elements
are not a finite linear
combination of them, but are a Hilbert basis.
