Definition:Big-O Notation/General Definition/Infinity

Definition
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.

Also see

 * Definition:Alexandroff Extension