 Solution

A polynomial $p_n\in {\Pi }_n$ is a best polynomial least-squares approximation when it minimizes the weighted error norm over all polynomials of degree at most $n$:Equivalently it minimizes ${\int }_0^1{x}^{-1/2}(f(x)-q(x){)}^{2}dx$. The weight is integrable, so the expression is finite for continuous functions; it defines a genuine norm because a continuous function with zero weighted squared integral must be identically zero.