Definition:Space of Convergent Sequences
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Definition
The space of convergent sequences, denoted $c$ is defined as:
- $\displaystyle c := \set{\sequence{z_n}_{n \in \N} \in \C^\N : \exists L \in \C : \forall \epsilon \in \R_{>0} : \exists N \in \R: n > N \implies \cmod {z_n - L} < \epsilon}$
As such, $c$ is a subspace of $\C^\N$, the space of all complex sequences.
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Also denoted as
The space of convergent sequences
- $c$
can be seen written as:
- $c_\infty$
Also see
- Definition:Convergent Sequence
- Definition:Space of Zero-Limit Sequences
- Definition:Space of Almost-Zero Sequences
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $1.1$: Normed and Banach spaces. Vector Spaces