Definition:Big-O Notation/General Definition/Point

Definition
Let $X$ 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.

Let $x_0\in X$.

The statement
 * $f(x) = \mathcal O \left({g(x)}\right)$ as $x\to x_0$

is equivalent to the statement:
 * There exists a neighborhood $U$ of $x_0$, and there exists $c\in \R: c\ge 0$ such that $\Vert f(x)\Vert\leq c\cdot\Vert g(x)\Vert$ for all $x\in U$