Definition:Space of Bounded Sequences/Vector Space

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

Let $\map {\ell^\infty} {\mathbb F}$ be the space of $\mathbb F$-valued bounded sequences. 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 $\mathbb F$.

Also see

 * Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space shows that $\struct {\map {\ell^\infty} {\mathbb F}, +, \circ}_{\mathbb F}$ is indeed a vector space.