Category:Space of Bounded Sequences

This category contains results about Space of Bounded Sequences.
Definitions specific to this category can be found in Definitions/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.

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

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 $\mathbb F$.

Subcategories

This category has only the following subcategory.

The following 2 pages are in this category, out of 2 total.