Definition:Big-O Notation/General Definition

Definition
Let $(X, \tau)$ be a topological space.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$

Let $f,g : X \to V$ be functions.

The statement:
 * $f(x) = \mathcal O \left({g(x)}\right)$ as $x \to \infty$

is equivalent to:
 * There exists a neighborhood of infinity $U \subset X$ such that:
 * $\displaystyle \exists c \in \R: c \ge 0 : \forall x \in U : \Vert f(x) \Vert \leq c \cdot \Vert g(x) \Vert$

That is, $\Vert f(x) \Vert \leq c \cdot \Vert g(x) \Vert$ for all $x$ in a neighborhood of infinity.

Also see

 * Definition:Alexandroff Extension