basic properties of seminorms

Proposition 1.

Suppose $p\mathrm{:}V\mathrm{\to }\mathrm{R}$ is a seminorm on a real (or complex) vector space $V$.Then

1. $p(0)=0$,

2. $p(v)\ge 0$ for all $v\in V$.

Proof.

Property $1$ follows using homogeneity;

$$p(0)=p(0\cdot 0)=|0|p(0)=0.$$

Property $2$ follows using sublinearity and Property 1;

$$0=p(0)=p(v-v)\le p(v)+p(-v)=2p(v).$$

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