\begin{definition}[Definition:O Notation/Big-O Notation/General Definition/Infinity]
Let $\struct {X, \tau}$ be a topological space.
Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$.
Let $f, g : X \to V$ be functions.
The statement:
:$\map f x = \map \OO {\map g x}$ as $x \to \infty$
is equivalent to:
:There exists a neighborhood of infinity $U \subset X$ such that:
::$\exists c \in {\R}_{\ge 0}: \forall x \in U: \norm {\map f x} \le c \norm {\map g x}$
That is:
:$\norm {\map f x} \le c \norm {\map g x}$
for all $x$ in a neighborhood of infinity.
\end{definition}