# 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.

## 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