Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$. For all functions $f$ and $g$ in $E$, we set
$$\langle f, g \rangle = \int_I f(x) g(x) w(x)\,\mathrm{d}x.$$
Show that we thus define an inner product on $E$.