Definition:O Notation/Little-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 $\norm {\,\cdot\,}$.

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

Let $x_0 \in X$.

The statement
 * $\map f x = \map o {\map g x}$ as $x \to x_0$

is equivalent to the statement:
 * For all $\epsilon > 0$, there exists a neighborhood $U$ of $x_0$ such that $\norm {\map f x} \le \epsilon \cdot \norm {\map g x}$ for all $x \in U$