Show that the norm
$$\|(x_{1},x_{2},...,x_{n})\|=|x_{1}|+|x_{2}|+...+|x_{n}|$$
can not be defined by a scalar product.
This question is part of a series of questions to do with how a scalar product defines a norm and the parallelogram identity. Given that context and some research I have concluded that any norm obtained from an inner product space should satisfy the parallelogram law. I am however stuck on how to proceed with question?
Hint. For $n\geq 2$, take $\mathbf{x}=(1,0,\dots,0)$ and $\mathbf{y}=(0,\dots,0,1)$. Is it true that $$\|\mathbf{x}+\mathbf{y}\|^2+\|\mathbf{x}-\mathbf{y}\|^2=2\|\mathbf{x}\|^2+2\|\mathbf{y}\|^2\quad ?$$