# Definition:Space of Bounded Sequences

## Definition

Let $\mathbb F \in \set {\R, \C}$.

We define the **space of bounded sequences** on $\mathbb F$, written $\map {\ell^\infty} {\mathbb F}$, by:

\(\ds \map {\ell^\infty} {\mathbb F}\) | \(=\) | \(\ds \set {\sequence {x_n}_{n \mathop \in \N} \in {\mathbb F}^\N : \sequence {x_n}_{n \mathop \in \N} \text { is a bounded sequence} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \set {\sequence {x_n}_{n \mathop \in \N} \in {\mathbb F}^\N : \sup_{n \mathop \in \N} \cmod {x_n} < \infty}\) |

where ${\mathbb F}^\N$ is the space of all $\mathbb F$-valued sequences.

### Vector Space

Let $+$ denote pointwise addition on the ring of sequences.

Let $\circ$ denote pointwise scalar multiplication on the ring of sequences.

We say that $\struct {\map {\ell^\infty} {\mathbb F}, +, \circ}_{\mathbb F}$ is the **vector space of bounded sequences** on $\mathbb F$.

### Normed Vector Space

Let $\norm \cdot_\infty$ be the supremum norm on the space of bounded sequences.

We say that $\struct {\map {\ell^\infty} {\mathbb F}, \norm \cdot_\infty}$ is the **normed vector space of bounded sequences** on $\mathbb F$.

## 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**

- $\map {\ell^\infty} {\mathbb F}$

can be seen written as:

- $\map {c_b} {\mathbb F}$

Where the field $\mathbb F$ can be easily inferred, we may simply write $\ell^\infty$ or $c_b$.

## Also see

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $1.1$: Normed and Banach spaces. Vector Spaces