Definition:Space of Bounded Sequences
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Definition
The space of bounded sequences, denoted $\ell^\infty$ is defined as:
- $\displaystyle \ell^\infty := \set{\sequence{z_n}_{n \mathop \in \N} \in \C^\N: \sup_{n \mathop \in \N} \size {z_n} < \infty}$
As such, $\ell^\infty$ is a subspace of $\C^\N$, the space of all complex sequences.
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Also known as
The space of bounded sequences is also known as space of absolutely bounded sequences.
Also denoted as
The space of bounded sequences
- $\ell^\infty$
can be seen written as:
- $c_b$
Also see
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $1.1$: Normed and Banach spaces. Vector Spaces