Definition:Limit Point/Normed Vector Space

Definition

Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $Y \subseteq X$ be a subset of $X$.

Limit Point of Set

Let $\alpha \in X$.

Then $\alpha$ is a limit point of $Y$ if and only if every deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $Y$:

$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap Y \ne \O$

that is:

$\forall \epsilon \in \R_{>0}: \set {x \in Y: 0 < \norm {x - \alpha} < \epsilon} \ne \O$

Note that $\alpha$ does not have to be an element of $A$ to be a limit point.

Limit Point of Sequence

Let $L \in Y$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $Y \setminus \set L$.

Let $\sequence {x_n}_{n \mathop \in \N}$ converge to $L$.

Then $L$ is a limit of $\sequence {x_n}_{n \mathop \in \N}$ as $n$ tends to infinity which is usually written:

$\ds L = \lim_{n \mathop \to \infty} x_n$