Biorthogonal system
A pair of sets and
,
, of elements of a (topological) vector space
and the dual (topological) space
, respectively, which satisfies the conditions
![]() |
if , and
if
(here,
is the canonical bilinear form coupling
and
). For instance, a biorthogonal system consists of a Schauder basis and the set formed by the expansion coefficients of
in it. In a Hilbert space
with scalar product
and basis
the set
satisfying the condition
![]() |
where if
and
if
, is also a basis; it is said to be the basis dual to
and, since
, the sets
and
form a biorthogonal system. In particular, a basis in
is said to be orthonormal if its dual to itself.
However, there also exist biorthogonal systems which do not even form a weak basis; an example is the set of functions ,
,
, in the space of continuous periodic functions with the norm
.
Biorthogonal system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Biorthogonal_system&oldid=11290