Let $E$ be a finite dimensional $\mathbb{K}$-vectorspace. Is $\|\cdot\|$ an arbitrary norm on $E$, then is every seminorm $p: E\to [0,\infty)$ continuous regarding $\|\cdot\|$.
I want to show this statement, but I am not sure what I have to do. I know what I have to do, when I have a system of seminorms:
Let $P$ be a system of seminorms on the $\mathbb{K}$-vectorspace $E$. Is $q: E\to [0,\infty)$ an arbitrary seminorm on $E$ then is $q$ continuous regarding $P$, if $\exists c\geq 0$ and $p_1,\dotso, p_n\in P$ with $q(x)\leq c\max\{p_1(x),\dotso, p_n(x)\}$ for all $x\in E$.
But now I do not have systems of seminorms. Can I do it similar?
So I have to find $c\geq 0$ such that $q(x)\leq c\|x\|$?
I would apprechiate just a clearification on what is to show, so I can try it myself first.
As a hint it is given to use, that norms are equivalent.
Thanks.
Let $e_1\dots e_n$ be a basis of $E$ such that $\|e_i\|=1$ for all $i=1\dots n$. Let $x\in E$ with $x=\sum_{i=1}^n a_ie_i$. Then we get $$ p(x) = p(\sum_{i=1}^n a_ie_i) \le \sum_{i=1}^n |a_i| p(e_i) \le \max_{i=1\dots n}p(e_i) \cdot \sum_{i=1}^n |a_i|. $$ The coordinate mapping $x\mapsto a_i$ is continuous from $(E,\|\cdot\|)$ to $\mathbb K$. Hence, there is a constant $c>0$ such that $|a_i|\le c \|x\|$.
This proves $$p(x) \le (c n \max_{i=1\dots n}p(e_i) )\cdot \|x\|.$$ The term in brackets is independent of $x$. Hence we obtain $$ p(x-y) \le (c n \max_{i=1\dots n}p(e_i) )\cdot \|x-y\| $$ and the mapping $x\mapsto p(x)$ is is continuous from $(E,\|\cdot\|)$ to $\mathbb R$.