In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$.\\
Show that we define an inner product on $E_1$ by setting
$$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$$