# True or False: Inner product on $\mathbb{R}^2$ satisfying a specific norm.

Verify or refute: There exists an inner product in $$\mathbb{R}^2$$ such that the norm of every vector $$v=(v_1,v_2)$$ is $$\|v\|=|v_1|+|v_2|$$.

I think this is untrue. So I took $$v=(1,0), y=(0,1)$$. After some calculations, I got that:

$$\|v+y\|^2-\|v-y\|^2= 4-4=0$$

and $$2\|v\|^2+ 2\|y\|^2=4$$.

But this clearly satisfies the parallelogram law since $$0\leq4$$.

So, does such an inner product exist? Or perhaps it doesn't exist but my election of vectors $$v,y$$ wasn't the appropriate?

• I am not sure what you mean by the parallelogram law, but for inner product spaces the parallelogram identity must hold (math.stackexchange.com/questions/21792/…), which is not the case in the example you have picked. Thus, this does indeed not yield an inner product space. Commented May 17 at 18:52
• Got it. Thanks! My confusion stems because I was working with an inequality "version" of the parallelogram law. Commented May 17 at 19:00

$$\|v+y\|^2+\|v-y\|^2=2\|v\|^2+ 2\|y\|^2$$