\begin{definition}[Definition:O Notation/Big-O Notation/Parameter]
{{refactor|Is this really a definition page? If so, then what exactly is it defining? Removed transclusion from its parent page while this is being clarified (by doing some research).}}
Let $X$ be a topological space.
Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\, \cdot \,}$
Let $x_0 \in X \cup \set \infty$.
Let $A$ be a set.
Let $X_\alpha$ be a subset of $X$ for every $\alpha \in A$, which, if $x_0 \ne \infty$, contains $x_0$.
Let $f_\alpha : X_\alpha \setminus \set {x_0} \to V$ be a mapping for every $\alpha \in A$.
Let $g : X \to V$ be a mapping.
The statement:
:$f_\alpha = \map {\OO_\alpha} g$ as $x \to x_0$
is equivalent to:
:$\forall \alpha \in A : f_\alpha = \map \OO g$ as $x \to x_0$
The $\OO$-estimate is said to be '''independent''' of $\alpha \in A$ {{iff}}:
:there exists a neighborhood $U$ of $x_0$ in $X$ such that:
::$\exists c \in \R: c \ge 0 : \forall \alpha \in A : \forall x \in \paren {U \setminus \set {x_0} } \cap X_\alpha : \norm {\map {f_\alpha} x} \le c \cdot \norm {\map g x}$
That is, if the implied constant and implied neighborhood can be chosen the same for all $\alpha \in A$.
Category:Definitions/Asymptotic Notation
Category:Definitions/Order Notation
\end{definition}