Riesz dual basis in an inner product space ($(u_i,{\hat{u}}_k)={\delta }_{ik}$)

For a basis $(u_i)$ of a finite-dimensional inner-product space with conjugate-linear first argument, define $G_{ij}=(u_i,u_j)$ and ${\hat{u}}_k={\sum }_l(G^{-1}{)}_{lk}{u}_l$. Then $(u_i,{\hat{u}}_k)={\delta }_{ik}$ and $({\hat{u}}_j,{\hat{u}}_k)=(G^{-1}{)}_{jk}$. This is the inner-product representation of the dual basis; requiring the dual vectors to lie in the given span is essential.