We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$, which is an antiderivative of $x \mapsto f ( x ) \frac { \mathrm { e } ^ { - x } } { x }$. The limits of $t \mapsto F(t)U(g)(t)$ at $0$ and $+\infty$ are both $0$. Show that
$$\langle f \mid U ( g ) \rangle = \int _ { 0 } ^ { + \infty } U ( f ) ^ { \prime } ( t ) U ( g ) ^ { \prime } ( t ) \mathrm { e } ^ { - t } \mathrm {~d} t.$$