Past exam of the options II course of the University of Oxford / 2010 / P / P3 / a

A best approximation to $x$ from $U_n$ is a vector $u_{\ast }\in U_n$ such thatThe definition requires attainment of the infimum. This finite-dimensional subspace does have such an approximation, as the explicit projection below shows; completeness of the whole inner-product space is unnecessary.