We equip $\mathbb{R}_{n}[X]$ with the inner product defined by
$$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$
and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$.
For $j \in \mathbb{N}$, we define the polynomial
$$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$
By convention, $P_{0} = 1$.
By means of integration by parts, show that the family $\left(P_{j}\right)_{0 \leqslant j \leqslant n}$ is orthogonal in $\mathbb{R}_{n}[X]$.