We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$, and the space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $\varphi(f,g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
a) Show that there exists a sequence of polynomial functions $(p_n)_{n \in \mathbb{N}}$ such that, for all $n \in \mathbb{N}$, $p_n$ has degree $n$ and leading coefficient 1, and that, for all $n \in \mathbb{N}^*$, $p_n$ is orthogonal to all elements of $E_{n-1}$.
b) Show that there exists a unique family $(Q_n)_{n \in \mathbb{N}}$ of polynomial functions satisfying the following conditions:
- [i)] the family $(Q_n)_{n \in \mathbb{N}}$ is orthogonal for the inner product $(\cdot|\cdot)$;
- [ii)] for all $n \in \mathbb{N}$, $Q_n$ has degree $n$ and leading coefficient 1.