How do I find a constant $B > 0 $, so that $\left\lVert x \right\rVert$ $\leq$ $ B* \left\lVert x \right\rVert_\infty$ works for all $x \in \mathbb{R}^n$?
I am not sure but I think I have to look at $x$ = $\sum_{i=1}^{n} x_ie_i$ with the development of the x to the canonical unit vectors $e_i$. But how do I do that formally?
Write $x = x_1 e_1 + \cdots + x_n e_n$. If $\|\cdot\|$ is an arbitrary norm on $\mathbb R^n$ you have $$\|x\| = \|x_1e_1 + \ldots x_n e_n\| \le \sum_{i=1}^n \|x_i e_i\| = \sum_{i=1}^n |x_i| \|e_i\|.$$ Each $x_i$ satisfies $|x_i| \le \|x\|_\infty$ so that $$\|x\| = \|x_1e_1 + \ldots x_n e_n\| \le \sum_{i=1}^n \|x\|_\infty \|e_i\| = \left(\sum_{i=1}^n \|e_i\| \right) \|x\|_\infty. $$ Take $$B = \sum_{i=1}^n \|e_i\|.$$