Definition:O Notation/Big-O Notation/Parameter

Definition
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 $\alpha \in A$ :
 * 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 $\alpha \in A$.