Let $\pi: E \to X$ be a smooth complex vector bundle of rank $r \in \mathbb{N}$ over a paracompact smooth manifold $X$. Let $\operatorname{Herm}(E) \to X$ denote the smooth real vector bundle whose fiber over $x \in X$ is the real [vector space](/page/Vector%20Space) of Hermitian forms $E_x \times E_x \to \mathbb{C}$. Then there exists a smooth section $h: X \to \operatorname{Herm}(E)$ such that, for every $x \in X$, the fiber map $h_x: E_x \times E_x \to \mathbb{C}$ is a positive-definite Hermitian form.