A complete biorthogonal system in a Banach space 
over 
or 
is a pair of indexed families 
in 
and 
in its dual
normed space 
. The vector space span
of the vectors 
is dense in 
, and
 |
(1)
|
where 
is the Kronecker delta. The elements of 
are continuous linear
functionals (Heil 1998, p. 50; Halbeisen 2003, p. 162).
Biorthogonality is a pairing between vectors and linear functionals. It does not require two families of functions to be mutually orthogonal.
For a sequence of such pairs, completeness alone does
not guarantee the norm-convergent expansion 
for every 
. That additional property makes 
a Schauder basis (Heil
1998, p. 51).
For example, in the L2-space ![L^2([-pi,pi])](/images/equations/CompleteBiorthogonalSystem/Inline15.svg)
,
take 
,
,
and 
for 
.
The corresponding continuous linear
functionals are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The orthogonality of the entire trigonometric family, including its constant member, gives 
. Its vector
space span is dense in the L2-space,
so these pairs form a complete biorthogonal system (Axler 2026, pp. 237 and
356). They supply the coefficient pairing for the Fourier
series.
