Space of Bounded Sequences with Supremum Norm forms Normed Vector Space
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Theorem
The vector space of bounded sequences with the supremum norm forms a normed vector space.
Proof
We have that:
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