$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients and we use the transformation $L$ applied to elements of $\mathcal{P}$ for the study of an operator $U$.
Let $P$ and $Q$ be two elements of $\mathcal{P}$.
Show that the integral $\displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, where $\bar{P}$ is the polynomial whose coefficients are the conjugates of those of $P$, converges.