Let $M$ be a paracompact smooth manifold, and let $\pi: E \to M$ be a smooth real vector bundle of finite rank $k \in \mathbb{N}$. Then there exists a smooth section $g \in \Gamma(\operatorname{Sym}^2 E^*)$ such that, for every $p \in M$, the [bilinear form](/page/Bilinear%20Form) $g_p: E_p \times E_p \to \mathbb{R}$ is an [inner product](/page/Inner%20Product) on the fibre $E_p$.