Definition:Space of Bounded Sequences/Vector Space
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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.