Definition:Limit of Function (Normed Vector Space)

Definition
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
 * $\displaystyle \lim_{x \mathop \to c} \map f x = L$

the following equivalent conditions hold:

$\epsilon$-Ball Condition
This is voiced:


 * the limit of $\map f x$ as $x$ tends to $c$.

Equivalence of Definitions
These definitions are seen to be equivalent in Equivalence of Definitions of Limit of Function in Normed Vector Space.

Also known as

 * $\map f x$ tends to the limit $L$ as $x$ tends to $c$

can also be voiced as:
 * $\map f x$ approaches the limit $L$ as $x$ approaches $c$