Past exam of the numerical analysis course of the University of Oxford / 2014 / 2 / a / Solution

The norm induced by the specified inner product is the L2 normA polynomial $p_n\in {\Pi }_n$ is a best polynomial least-squares approximation whenMinimizing this norm is equivalent to minimizing its square, the integrated squared error. Continuity ensures that a function with zero squared norm is identically zero: a nonzero value would give a neighbourhood on which the square has a positive integral.