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=31234