= Riesz dual basis in an inner product space
{c}
{title2=$(u_i,\widehat u_k)=\delta_{ik}$}
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 $\widehat u_k=\sum_l(G^{-1})_{lk}u_l$. Then $(u_i,\widehat u_k)=\delta_{ik}$ and $(\widehat u_j,\widehat u_k)=(G^{-1})_{jk}$. This is the inner-product representation of the <dual basis>; requiring the dual vectors to lie in the given span is essential.
Back to article page
