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:

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


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

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

 * Definition:Supremum Norm
 * Definition:P-Sequence Space
 * Definition:Bounded Sequence