How to disprove that there is an inner product on $\mathbb{R^2}$ s.t. the norm is $||(x_1,x_2)|| = |x_1|+|x_2|$? How can I disprove that there is an inner product on $\mathbb{R^2}$ s.t. the norm is $||(x_1,x_2)|| = |x_1|+|x_2|$?
My approach is to use the parallelogram law to show that if I have two vectors $u,w \in \mathbb{R^2}$, then if it ISNT a norm, 
$||u+w||^2+||u-w||^2 \neq 2(||u||^2+||w||^2)$. 
However, I dont understand why this parallelogram law would show anything as the theorem in my book didnt mentioned anything about the parallelogram law and norms. Thanks!

If the norm is induced by an inner product, parallelogram law necessarily holds, since
$$
\left\|u+w\right\|^2+\left\|u-w\right\|^2=\left<u+w, u+w\right> + \left<u-w, u-w\right>=\\
= \left<u, u\right> + 2\left<u, w\right>+\left<w, w\right> + \left<u, u\right> - 2\left<u, w\right>+\left<w, w\right> =\\
\\
=2\left(\left<u, u\right>+\left<w, w\right>\right)=2\left(\left\|u\right\|^2+\left\|w\right\|^2\right)
$$
As it does not hold for the norm in question, it cannot be induced by an inner product.
Example: $u=(1,1)$, $w=(0,3)$.
