We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form
$$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$
where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set
$$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f: t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g: t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$
Show that $E_2$ equipped with $\langle \cdot, \cdot \rangle$ is a real pre-Hilbert space.