Biorthogonal system

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

:$ilde\; v\_i$ in "E" and $ilde\; u\_i$ in "F"

such that

:$langle\; ilde\; v\_i\; ,\; ilde\; u\_j\; angle\; =\; delta\_\{i,j\}$

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 "L"² [0,2π] the functions cos "nx" and sin "nx" form a biorthogonal system.

Projection

Related to a biorthogonal system is the projection :$P:=\; sum\_\{i\; in\; I\}\; ilde\; u\_i\; otimes\; ilde\; v\_i$,where $left(\; u\; otimes\; v\; ight)\; (x):=\; u\; langle\; v,\; x\; angle$; its image is the linear span of $\{\; ilde\; u\_i:\; i\; in\; I\}$, and the
kernel is $\{langle\; ilde\; v\_i,\; cdot\; angle\; =\; 0:\; i\; in\; I\; \}$.

Construction

Given a possibly non-orthogonal set of vectors $mathbf\{u\}=\; (u\_i)$ and $mathbf\{v\}=\; (v\_i)$ the projection related is :$P=\; sum\_\{i,j\}\; u\_i\; left(\; langlemathbf\{v\},\; mathbf\{u\}\; angle^\{-1\}\; ight)\_\{j,i\}\; otimes\; v\_j$, where $langlemathbf\{v\},mathbf\{u\}\; angle$ is the matrix with entries $left(langlemathbf\{v\},mathbf\{u\}\; angle\; ight)\_\{i,j\}=\; langle\; v\_i,\; u\_j\; angle$.
* $ilde\; u\_i:=\; (I-P)\; u\_i$, and $ilde\; v\_i:=\; left(I-P\; ight)^*\; v\_i$ then is an orthogonal system.

ee also

* Dual space
* Dual pair
* Orthogonality
* Orthogonalization

References

*Jean Dieudonné, "On biorthogonal systems" Michigan Math. J. 2 (1953), no. 1, 7–20 [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1028989861]