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]