For all $f \in E$, we set,
$$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$
where $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$
For all $f \in E$, show that $T(f)$ is of class $\mathcal{C}^2$ then that $T(f)'' = -f$.