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 $X$.

Proof

Seminorm Axiom $\text N 2$: Positive Homogeneity

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

We have:

\(\ds \map q {k x}\)

\(=\)

\(\ds \alpha \map p {k x}\)

\(\ds \)

\(=\)

\(\ds \alpha \cmod k \map p x\)

Seminorm Axiom $\text N 2$: Positive Homogeneity for $p$

\(\ds \)

\(=\)

\(\ds \cmod k \map q x\)

$\Box$

Seminorm Axiom $\text N 3$: Triangle Inequality

Let $x, y \in X$.

Then we have:

\(\ds \map q {x + y}\)

\(=\)

\(\ds \alpha \map p {x + y}\)

\(\ds \)

\(\le\)

\(\ds \alpha \paren {\map p x + \map p y}\)

Seminorm Axiom $\text N 3$: Triangle Inequality for $p$

\(\ds \)

\(=\)

\(\ds \alpha \map p x + \alpha \map p y\)

\(\ds \)

\(=\)

\(\ds \map q x + \map q y\)

$\blacksquare$