Absolute Value is Norm

Theorem

The absolute value is a norm on the set of real numbers $\R$.

Proof

By Complex Modulus is Norm then the complex modulus satisfies the norm axioms on the set of complex numbers $\C$.

Since the real numbers $\R$ is a subset of the complex numbers $\C$ then the complex modulus satisfies the norm axioms on the real numbers $\R$.

By Complex Modulus of Real Number equals Absolute Value then the absolute value satisfies the norm axioms on set of real numbers $\R$.

$\blacksquare$

Sources

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