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

Given a possibly non-orthogonal set of vectors ${\displaystyle \mathbf {u} =\left(u_{i}\right)}$  and ${\displaystyle \mathbf {v} =\left(v_{i}\right)}$  the projection related is $${\displaystyle P=\sum _{i,j}u_{i}\left(\langle \mathbf {v} ,\mathbf {u} \rangle ^{-1}\right)_{j,i}\otimes v_{j},}$$  where ${\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle }$  is the matrix with entries ${\displaystyle \left(\langle \mathbf {v} ,\mathbf {u} \rangle \right)_{i,j}=\left\langle v_{i},u_{j}\right\rangle .}$ 

• ${\displaystyle {\tilde {u}}_{i}:=(I-P)u_{i},}$  and ${\displaystyle {\tilde {v}}_{i}:=(I-P)^{*}v_{i}}$  then is a biorthogonal system.

• Dual basis – Linear algebra concept
• Dual space – In mathematics, vector space of linear forms
• Dual pair
• Orthogonality – Various meanings of the terms
• Orthogonalization

• Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [1]