Non-Negative Scalar Multiple of Seminorm on Vector Space is Seminorm

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

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

Let $p$ be a seminorm on $X$.

Let $\alpha \in \R_{\ge 0}$.

Let $q = \alpha p$.

Then $q$ is a seminorm on $X$.

Let $x \in X$ and $k \in \GF$.

We have:

Let $x, y \in X$.

Then we have: