\begin{definition}[Definition:O Notation/Big-O Notation/Uniform]
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''' {{iff}}:
:$\exists c > 0 : \forall x \in X : \norm {\map f x} \le c \cdot \norm {\map g x}$
This is denoted:
:$f = \map \OO g$
Category:Definitions/Asymptotic Notation
\end{definition}