Definition:Big-O Notation/Uniform

Definition
Let $X$ be a set.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$.

Let $f,g : X \to V$ be mappings.

Then $f$ is big O of $g$ uniformly :
 * $\exists c>0 : \forall x \in X : \Vert f(x) \Vert \leq c \cdot \Vert g(x) \Vert$

This is denoted: $f=O(g)$.