Definition:Space of Convergent Sequences

Definition

The space of convergent sequences, denoted $c$ is defined as:

$\ds c := \set {\sequence{z_n}_{n \mathop \in \N} \in \C^\N : \exists L \in \C : \forall \epsilon \in \R_{>0} : \exists N \in \R: n > N \implies \cmod {z_n - L} < \epsilon}$

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