Space of Bounded Sequences with Supremum Norm forms Normed Vector Space

Theorem

The vector space of bounded sequences with the supremum norm forms a normed vector space.

Proof

We have that:

Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space

Supremum norm on the space of bounded sequences is a norm

By definition, $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ is a normed vector space.

$\blacksquare$

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces