# Category:Definitions/Space of Bounded Sequences

This category contains definitions related to Space of Bounded Sequences.

Related results can be found in Category:Space of Bounded Sequences.

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 $\mathbb F$.

## Pages in category "Definitions/Space of Bounded Sequences"

The following 4 pages are in this category, out of 4 total.