Definition:Convergent Sequence/Normed Vector Space
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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