\begin{definition}[Definition:Limit of Function (Normed Vector Space)/Epsilon-Delta Condition]
Let $M_1 = \struct {X_1, \norm {\,\cdot\,}_{X_1}}$ and $M_2 = \struct {X_2, \norm {\,\cdot\,}_{X_2}}$ be normed vector spaces.
Let $c$ be a limit point of $M_1$.
Let $f: X_1 \to X_2$ be a mapping from $X_1$ to $X_2$ defined everywhere on $X_1$ ''except possibly'' at $c$.
Let $L \in M_2$.
$\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $c$''' and is written:
:$\map f x \to L$ as $x \to c$
or
:$\ds \lim_{x \mathop \to c} \map f x = L$
{{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \norm {x - c}_{X_1} < \delta \implies \norm {\map f x - L}_{X_2} < \epsilon$
That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that ''every'' point in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some point $L$ in the codomain of $f$.
This is voiced:
: '''the limit of $\map f x$ as $x$ tends to $c$'''.
\end{definition}