Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$:
$$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$
We denote by $T$ the application defined on $E$ such that:
$$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$
Show that $T \in \mathcal{L}(E)$ with this norm.