Past exam of the mathematical methods I course of the University of Oxford / 2018 / 10 / b / Solution

Expand the vector in the orthonormal basis, $v={\sum }_j{v}_j{e}_j$. Using linearity in the second slot and the Kronecker delta givesExpanding both vectors and using conjugate linearity in the first slot therefore givesAn orthonormal set that is not a basis would recover only the coordinates of the orthogonal projection onto its linear span; here the $n$ vectors span ${\mathbb{C}}^n$.