I am trying to show the above by making use of the theorem that "A norm is induced by an inner product iff the Parallelogram law holds for this norm" where Parallelogram law $$\|x+y\|^2+\|x-y\|^2=2(\|x\|^2 + \|y\|^2)$$ but am not sure how to use it to apply it to the above question.
What would the $y$ variable be in my case?
Any help is welcome.
If that norm is induced by an inner product then, by the Parallelogram Law, for all ${\bf v_1}=(x_1,y_1)$ and ${\bf v_2}=(x_2,y_2)$, we have that $$\|{\bf v_1}+{\bf v_2}\|^2+\|{\bf v_1}-{\bf v_2}\|^2=2(\|{\bf v_1}\|^2 + \|{\bf v_2}\|^2)$$ that is $$(|x_1+x_2|+|y_1+y_2|)^2 + (|x_1-x_2|+|y_1-y_2|)^2 = 2(|x_1|+|y_1|)^2+2(|x_2|+|y_2|)^2.$$ Try to find two vectors $(x_1,y_1)$ and $(x_2,y_2)$ such that the equality does not hold.