We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$ Determine a family of polynomials $\left(K_n\right)_{n \in \mathbb{N}}$ satisfying the following two conditions: i. for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ and its leading coefficient is strictly positive; ii. for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for the inner product $\langle \cdot, \cdot \rangle$. Justify the uniqueness of such a family.
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by:
$$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$
Moreover, we set:
$$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Determine a family of polynomials $\left(K_n\right)_{n \in \mathbb{N}}$ satisfying the following two conditions:
i. for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ and its leading coefficient is strictly positive;
ii. for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for the inner product $\langle \cdot, \cdot \rangle$.
Justify the uniqueness of such a family.