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;
$$p(0)=p(0\cdot 0)=0p(0)=0.$$ 
Property $2$ follows using sublinearity and Property 1;
$$0=p(0)=p(vv)\le p(v)+p(v)=2p(v).$$ 
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Title  basic properties of seminorms 

Canonical name  BasicPropertiesOfSeminorms 
Date of creation  20130322 14:38:57 
Last modified on  20130322 14:38:57 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  5 
Author  matte (1858) 
Entry type  Theorem 
Classification  msc 46B20 