P-Sequence Space with P-Norm forms Normed Vector Space

Theorem

A $p$-Sequence Space with a $p$-norm forms a normed vector space.

Proof

We have that:

a $p$-sequence space is a vector space

the $p$-norm on the $p$-sequence space is a norm

By definition, $\struct {\ell^p, \norm {\, \cdot \,}_p}$ 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