In this part, $E$ denotes the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by,
$$\forall (f,g) \in E^2, \quad \langle f,g \rangle = \int_0^1 f(t)g(t)\,\mathrm{d}t$$
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 $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$
Show that $T$ is a continuous endomorphism of $E$.