(Biorthogonality) The collection of functions
is "biorthogonal" to the collection
iff
where the
is inner product defined by context
(
-inner
product for applications of this section).

Proposition

(Linear independence from
biorthogonality) Let
be a collection of functions. If there is a biorthogonal collection
then
are linearly independent.

Proof

Assume existence of numbers
such
that
Apply
for every
: