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 $$N(x)=0\Longrightarrow x=0$$.
$\endgroup$3 Answers
$\begingroup$Take $N(x,y) = |x|$ on $\mathbb{R}^2$.
$\endgroup$ 1 $\begingroup$There’s the trivial seminorm: $N(x)=0$ for all $x\in V$.
$\endgroup$ $\begingroup$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 $x=c_1e_1+c_2e_2$.
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