Definition:Big-O Notation/General Definition

Definition
Let $\left({X, \tau}\right)$ 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:
 * $f \left({x}\right) = \mathcal O \left({g \left({x}\right)}\right)$ as $x \to \infty$

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

That is:
 * $\Vert f \left({x}\right) \Vert \le c \cdot \Vert g \left({x}\right) \Vert$

for all $x$ in a neighborhood of infinity.

Also see

 * Definition:Alexandroff Extension