We are given a real $a > 0$. We consider the space $E_3$ of functions $f : [0,a] \rightarrow \mathbb{R}$, continuous and of class $\mathcal{C}^1$ piecewise on $[0,a]$, and satisfying $f(0) = 0$. We equip $E_3$ with the inner product defined, for $f, g \in E_3$, by
$$(f \mid g) = \int_0^a f'(t) g'(t) \, \mathrm{d}t$$
Show that the function $(x,y) \mapsto \min(x,y)$ is a reproducing kernel on $(E_3, (\cdot \mid \cdot))$.