Definition:Space of Bounded Sequences/Normed Vector Space

Definition

Let $\mathbb F \in \set {\R, \C}$.

Let $\map {\ell^\infty} {\mathbb F}$ be 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 $\mathbb F$.

Also see

• Space of Bounded Sequences with Supremum Norm forms Normed Vector Space shows that $\struct {\map {\ell^\infty} {\mathbb F}, \norm \cdot_\infty}$ is indeed a normed vector space.