Norm on Vector Space is Seminorm

Theorem

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $\norm {\, \cdot \,}$ be a norm on $X$.

Then $\norm {\, \cdot \,}$ is a seminorm.

Proof

Note that Norm Axiom $\text N 2$: Positive Homogeneity and Norm Axiom $\text N 3$: Triangle Inequality are precisely Seminorm Axiom $\text N 2$: Positive Homogeneity and Seminorm Axiom $\text N 3$: Triangle Inequality.

$\blacksquare$