We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$. The previous question shows that the following application $\varphi$ is well defined: $$\varphi : \left\{ \begin{aligned} E \times E & \rightarrow \mathbb{R} \\ (f, g) & \mapsto \int_{-1}^{1} \frac{1}{\sqrt{1 - x^2}} f(x) g(x)\, dx \end{aligned} \right.$$ Show that $\varphi$ defines an inner product on $E$.
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
The previous question shows that the following application $\varphi$ is well defined:
$$\varphi : \left\{ \begin{aligned} E \times E & \rightarrow \mathbb{R} \\ (f, g) & \mapsto \int_{-1}^{1} \frac{1}{\sqrt{1 - x^2}} f(x) g(x)\, dx \end{aligned} \right.$$
Show that $\varphi$ defines an inner product on $E$.