# Definition:Space of Bounded Sequences

## 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:

$c_b$