Given that $E$ is a finite dimensional space. Let $\dim E=n\geq 1$ and $\{e_i\}^{n}_{i=1}$ be a basis for $E.$ Then, there exists unique scalars $\{\alpha_i\}^{n}_{i=1}$ such that \begin{align}x=\sum_{i=1}^{n}\alpha_i e_i.\end{align} The problem is: I want to prove that $\| \cdot \|_0$ defined by $\| x \|_0=\max\limits_{1\leq i\leq n}|\alpha_i|$ is a norm on $E$.
I, therefore, post the proof in the answer section after it has been approved.