Consider a set of polynomials of degree 2 or lower with complex coefficients, $\mathbb{P}_2(\mathbb{C})$. On this space we define the inner product $$\langle p,q\rangle = p(0)\overline{q(0)} + \int_{0}^{1} p'(x)\overline{q'(x)}dx$$ we define the following operator $(S(p(x)) = p(-x)$, thus S mirrors the function in vertical axis. Is $S$ self-adjoint?
My intuition says no, but I cannot find a counter example, could someone help me out? Thanks!