projection of the span on orthonormal vectors

definition

• ${\min }_{x}f(x):={\Vert p-{\sum }_{i=1}^d{u}_i{x}_i\Vert }_2^2$, $p\in R^n,x\in R^d,\{u_1,\cdots ,u_d\}$ is a set of orthonormal vectors in $R^n$
• $\begin{array}{c}f(x)=\Vert p{\Vert }_2^2+{\left({\sum }_{i=1}^d{u}_i{x}_i\right)}^T\left({\sum }_{i=1}^d{u}_i{x}_i\right)-2p^T\left({\sum }_{i=1}^d{u}_i{x}_i\right)\\ =\Vert p{\Vert }_2^2+{\sum }_{i=1}^m(x_i^2-2xi(p^T{u}_i)).\end{array}$