Modulus of Linear Functional on Vector Space is Seminorm

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

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

Let $f : X \to \GF$ be a linear functional.

Define $p_f : X \to \R_{\ge 0}$ by:


 * $\map {p_f} x = \cmod {\map f x}$

for each $x \in X$.

Then $p_f$ is a seminorm.

Proof of
For each $\lambda \in \GF$ and $x \in X$, we have:

Proof of
For each $x, y \in X$, we have: