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 $n$. For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ and $T_0(x) = 1$. The space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $(f|g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
Calculate $(T_m | T_n)$ for all $(m, n) \in \mathbb{N} \times \mathbb{N}$. What can we deduce from this?