Definition:Cauchy Sequence/Normed Vector Space
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Definition
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space.
Let $\sequence {x_n}$ be a sequence in $V$.
Then $\sequence {x_n}$ is a Cauchy sequence if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$
Also see
Source of Name
This entry was named for Augustin Louis Cauchy.
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