Prove norm doesn't come from inner product.
Pick two vectors $x$, $y$ in the space under consideration, which is presumably $\mathbb{R}^2$.
Calculate $\|x+y\|$, $\|x-y\|$, $\|x\|$ and $\|y\|$ using the norm you are given, in this case $\|\cdot\|_1$.
See whether the parallelogram rule holds for this particular $x$, $y$.
If it fails, then you know that the norm does not come from an inner product.