An example for a seminorm on $\mathbb{R}^n$

Question

Can any one come up with an example of a seminorm that is not a norm on $\mathbb{R}^n$ ?

A seminorm on a real vector space $V$ is a function $N:V\rightarrow \mathbb{R}$ that satisfies that

1) $N(x)\geq 0$, $x\in V$

2) $N(\alpha x)=|\alpha|N(x)$, $x\in V$, $\alpha\in \mathbb{R}$

3) $N(x+y)\leq N(x)+N(y)$, $x,y\in V$

So a seminorm generalizes a norm as it does not require the condition $$N(x)=0\Longrightarrow x=0$$.

Answer 1

Take $N(x,y) = |x|$ on $\mathbb{R}^2$.

Answer 2

Another example, if ${e_1,e_2}$ is a basis on $\mathbb{R}^2$, then define $N(x) = |c_1+c_2|$, where $x\in \mathbb{R}^2$ has the unique linear combination representation $x=c_1e_1+c_2e_2$.

Answer 3

There’s the trivial seminorm: $N(x)=0$ for all $x\in V$.