Past exam of the options II course of the University of Oxford / 2013 / P3 / a / Solution

With the weighted inner product $⟨p,q{⟩}_w={\int }_{-1}^{1}p(x)q(x)w(x)dx$, a family of orthogonal polynomials has $\mathrm{deg}{p}_j=j$, each $p_j$ nonzero, andContinuity and positivity of the weight make the integral finite and give $⟨p_j,p_j{⟩}_w>0$. Unit norm or monic leading coefficient is an additional optional normalization.