Definition:Limit of Sequence/Normed Vector Space

Definition

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

Let $L \in X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

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:

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

Also known as

A limit of $\sequence {x_n}$ as $n$ tends to infinity can also be presented more tersely as a limit of $\sequence {x_n}$ or even just limit of $x_n$.

Some sources present $\ds \lim_{n \mathop \to \infty} x_n$ as $\lim_n x_n$.

Also see

• Results about limits of sequences can be found here.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces