Category:Definitions/Space of Bounded Sequences

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This category contains definitions related to Space of Bounded Sequences.
Related results can be found in Category: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 "Definitions/Space of Bounded Sequences"

The following 4 pages are in this category, out of 4 total.