Definition:Big-O Notation/Uniform

Definition

Let $X$ be a set.

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

Let $f, g : X \to V$ be mappings.

Then $f$ is big-$\OO$ of $g$ uniformly if and only if:

$\exists c > 0 : \forall x \in X : \norm {\map f x} \le c \cdot \norm {\map g x}$

This is denoted:

$f = \map \OO g$