Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / P / P3 / a / Solution

The best polynomial minimizes the weighted squared error among all polynomials of degree at most $n$:The given inner product is positive definite on continuous functions despite the weight vanishing at zero: a continuous nonzero function is nonzero on an interval containing nonzero points, so its weighted square integral is strictly positive.