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
- Results about the space of bounded sequences can be found here.
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $1.1$: Normed and Banach spaces. Vector Spaces