> [!theorem] > > Let $U \subset \real^n$ be an [[Open Set|open]] set, $g: U \to \symbi(\real^2)$ be a [[Riemannian Metric|Riemannian metric]], and $\ol g$ is the standard Euclidean metric. Let $K \subset U$ be [[Compactness|compact]], then there exists $c, C > 0$ such that > $ > c\abs{v}_{\ol g} \le \abs{v}_{g} \le C \abs{v}_{\ol{g}} > $ > In other words, the given metric is equivalent to the standard Euclidean metric. > > *Proof*. The high and low norms on $\symbi$ are both continuous. Since $g$ is also continuous, $g$ composed with both has a maximum and minimum, respectively. The minimum cannot be zero, because $g$ is positive definite.