Set of Points for which Seminorm is Zero is Vector Subspace

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

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

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

Let:


 * $U = \set {x \in X : \map p x = 0}$

Then $U$ is a vector subspace of $X$.

Proof
From Seminorm Maps Zero Vector to Zero, $\map p {\mathbf 0_X} = 0$.

So $\mathbf 0_X \in U$ and in particular $U \ne \O$.

So we look to apply One-Step Vector Subspace Test.

Let $x, y \in U$ and $\lambda, \mu \in \GF$.

Then we have:

Since $\map p {\lambda x + \mu y} \ge 0$, so:


 * $\map p {\lambda x + \mu y} = 0$

so $\lambda x + \mu y \in U$.

By One-Step Vector Subspace Test, we have that $U$ is a vector subspace of $X$.