Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We set, for all $f \in E_1$,
$$U(f)(s) = \int_0^1 k_s'(t) f'(t)\,\mathrm{d}t$$
where $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$
Let $f \in E_1$ of class $\mathcal{C}^2$. Show that $U(f) = -T(f'')$. Deduce that $U(f) = f$.