I have a problem where I have to show that every $n$-dimensional normed space $E$ has the same norm as the euclidean space $E_n$.
Here's what I've got:
Since $E$ is $n$-dimensional then for the basis $(e_1, ..., e_n)$ of $E$ there's a unique representation for every $x \in E$ i.e. $x=\sum_{i=1}^{n}{\lambda_i e_i}$.
First I show for every $x \in E$ the function $g(x)=(\lambda_1, ... \lambda_n) \in E_n$ is an isomorphism.
Then I show that $\lVert x\rVert_{E} \le C\lVert v\rVert_{E_n}$ and thus the image is continuous.
After that I show that$\lVert x\rVert_{E_n} \le C_2\lVert v\rVert_{E}$ and thus the image is homeomorphic.
I also conclude that $m\lVert v\rVert_{E_n} \le \lVert x\rVert_{E} \le M\lVert v\rVert_{E_n}$ for every $x$.
I'm not clear on how to conclude that the norms are equivalent. How can I use the last inequality to show that?
Thanks in advance!