A Hilbert basis of a Hilbert space 
is an orthonormal set 
whose vector
space span is dense in 
. Equivalently, it is an orthonormal
basis in the Hilbert-space sense. Every element 
has the norm-convergent
expansion
 |
where 
for the inner product linear in its first argument,
and 
(Axler 2026, pp. 242-244).
For an arbitrary index set 
, convergence means that for every 
some finite subset 
satisfies 
for every finite subset 
with 
(Axler 2026, p. 239). For each
, at most countably many coefficients 
are nonzero.
If 
is a separable
space, a Hilbert basis can be indexed by a finite set
or countable set. A nonseparable Hilbert
space requires an uncountable set of indices.
For example, in the Hilbert space 
of square summable sequences,
a Hilbert basis is the standard basis 
, whose 
th entry is 
, the Kronecker
delta. Then
 |
with 
. The 
do not form an algebraic vector
basis of 
, since some elements,
such as 
, are not finite linear
combinations of them (Axler 2026, p. 241).
