Definition:Space of Bounded Sequences

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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:

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

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