Question

Let \(w(x)\) be a positive continuous function on the interval \(0 \leq x \leq 1, n\) a positive integer, and \(\mathcal{P}_{n}\) the vector space of polynomials \(p(x)\) whose degrees are at most \(n\) equipped with the inner product
\[
\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x .
\]
a) Prove that \(\mathcal{P}_{n}\) has an orthonormal basis \(p_{0}, p_{1}, \ldots, p_{n}\) with the degree of \(p_{k}\) is \(k\) for each \(k\).
b) Prove that \(\left\langle p_{k}, p_{k}^{\prime}\right\rangle=0\) for each \(k\).