Let $\{p_n\}_{n\in\mathbb{N}}$ be a basis for the polynomials with degree at most $n-1$. We define the inner product $\langle p, q\rangle := \int\limits_0^{1} p(t)q(t)w(t)$, for a continous function $w$.
Is it true that $\langle p_k,p'_k\rangle = 0, \forall 0 \leq k \leq n$, for arbitrary choice of normalized (i.e. $ \| p_k\|= \sqrt{\langle p_k,p_k \rangle} = 1$) basis polynomials? If so, does this still hold for any orthogonal set?
If one or both are true, how could one proof that?