For $x = \left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}\,,\ n \geq 2\ $, define $\left\vert\,x\,\right\vert := \max_i \left\vert\,x_i\,\right\vert\ $. Show that there exists no inner product $\left\langle ,\right\rangle$ on $\mathbb{R}^{n},\ $ for which $\left\langle x,x \right\rangle = \left\vert\,x\,\right\vert^{2}\,, \forall\ x \in \mathbb{R}^{n}$.
I've been unable to find a contradiction, although I think that is key to solving this problem.