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.

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 $\mathbb F$.

Subcategories

This category has only the following subcategory.

Pages in category "Space of Bounded Sequences"

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