Definition:Convergent Sequence/Normed Vector Space

Definition

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

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

Let $L \in X$.

The sequence $\sequence {x_n}_{n \mathop \in \N}$ converges to the limit $L \in X$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - L} < \epsilon$

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