# Can the norm given by $\|x\| = |x_1| + |x_2|$ be induced by an inner product?

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.

• A norm is induced by an inner product if and only if it satisfies the parallelogram law. Your norm does not satisfy the parallelogram law, that should be easy to check. Commented Oct 21, 2016 at 8:39
• What would the y variable in the Parallelogram law be in my case, please? Commented Oct 21, 2016 at 8:56
• $x$ and $y$ are any two members of the normed space. For example, if you are norming real number pairs, then $x$ and $y$ are real number pairs. Similarly, if you are norming function pairs, then $x$ and $y$ are function pairs, and $x_1,x_2,y_1,y_2$ are real numbers/functions respectively. Commented Oct 21, 2016 at 9:00

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.