\begin{definition}[Definition:O Notation/Little-O Notation/General Definition/Point]
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 $x \in U$
Category:Definitions/Asymptotic Notation
Category:Definitions/Order Notation
\end{definition}