Real Numbers with Absolute Value form Normed Vector Space

Theorem

Let $\R$ be the set of real numbers.

Let $\size {\, \cdot \,}$ be the absolute value.

Then $\struct {\R, \size {\, \cdot \,}}$ is a normed vector space.

Proof

We have that:

Real Numbers form Vector Space

Absolute Value is Norm

By definition, $\struct {\R, \size {\, \cdot \,}}$ is a normed vector space.

$\blacksquare$

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed Spaces