Biorthogonal system

In mathematics, a biorthogonal system is a pair of indexed families of vectors $$\tilde v_i \text{ in } E \text{ and } \tilde u_i \text{ in } F$$ such that $$\left\langle\tilde v_i, \tilde u_j\right\rangle = \delta_{i,j},$$ where $$E$$ and $$F$$ form a pair of topological vector spaces that are in duality, $$\langle \,\cdot, \cdot\, \rangle$$ is a bilinear mapping and $$\delta_{i,j}$$ is the Kronecker delta.

An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct.

A biorthogonal system in which $$E = F$$ and $$\tilde v_i = \tilde u_i$$ is an orthonormal system.

Projection
Related to a biorthogonal system is the projection $$P := \sum_{i \in I} \tilde u_i \otimes \tilde v_i,$$ where $$(u \otimes v) (x) := u \langle v, x \rangle;$$ its image is the linear span of $$\left\{\tilde u_i: i \in I\right\},$$ and the kernel is $$\left\{\left\langle \tilde v_i, \cdot \right\rangle = 0 : i \in I\right\}.$$

Construction
Given a possibly non-orthogonal set of vectors $$\mathbf{u} = \left(u_i\right)$$ and $$\mathbf{v} = \left(v_i\right)$$ the projection related is $$P = \sum_{i,j} u_i \left(\langle\mathbf{v}, \mathbf{u}\rangle^{-1}\right)_{j,i} \otimes v_j,$$ where $$ \langle\mathbf{v},\mathbf{u}\rangle $$ is the matrix with entries $$\left(\langle\mathbf{v}, \mathbf{u}\rangle\right)_{i,j} = \left\langle v_i, u_j\right\rangle.$$
 * $$\tilde u_i := (I - P) u_i,$$ and $$\tilde v_i := (I - P)^* v_i$$ then is a biorthogonal system.