Definition:Limit Point/Normed Vector Space/Sequence

Definition

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

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

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:

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

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $1$: Normed and Banach spaces