Biorthogonal system

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In mathematics, a biorthogonal system in a pair of topological vector spaces E and F that are in duality is a pair of indexed subsets

$\tilde v_i$ in E and $\tilde u_i$ in F

such that

$<\tilde v_i | \tilde u_j>= \delta_{i,j}$

with the Kronecker delta. This applies, for example, with E = F = H a Hilbert space; in which case this reduces to an orthonormal system. In L²[0,2π] the functions cos nx and sin nx form a biorthogonal system.

1 Projection
2 Construction
3 See also
4 Reference

[edit] Projection

Related to a biorthogonal system is the projection

$P:= \sum_{i \in I} \tilde u_i \otimes \tilde v_i$ ,

where $\left( u \otimes v\right) (x):= u <v|x>$ ; its image is the linear span of $\{\tilde u_i: i \in I\}$ , and the kernel is $\{<\tilde v_i|.>=0: i \in I \}$ .