- in E and in F
A biorthogonal system in which E = F is an orthonormal system.
In L2 [0, 2π] the functions cos(nx) and sin(nx) form a biorthogonal system. Another example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue.
Related to a biorthogonal system is the projection
Given a possibly non-orthogonal set of vectors and the projection related is
where is the matrix with entries .
- , and then is an orthogonal system.
See also 
- Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20