Past exam of the linear algebra course of the University of Oxford / 2017 / 3 / b / i / Solution

The integral is conjugate-linear in $f$, linear in $g$, and conjugate-symmetric, by the corresponding properties of the integral and of complex conjugation. Also,If a polynomial $f$ is not zero, there is a real point $x_0$ with $f(x_0)\ne 0$: a nonzero polynomial cannot vanish at every real point. By continuity, $|f|$ has a positive lower bound on a sufficiently small closed interval about $x_0$. The continuous function $e^{-x^2}$ also has a positive lower bound there. The integral over this interval is strictly positive. HenceTogether with the finiteness allowed in the question, these properties prove that the formula defines an inner product on $V=\mathbb{C}[x]$.