Sum norm and inner product: ||ax^2+bx^2+c||_s = sqrt{<ax^2+

[JellyFish]

I need to show that the sum norm on P[sub:1epcx0w4]2[/sub:1epcx0w4] is not defined form an inner product. So show that there is no inner product <,> on P[sub:1epcx0w4]2[/sub:1epcx0w4] such that
||ax[sup:1epcx0w4]2[/sup:1epcx0w4] +bx[sup:1epcx0w4]2[/sup:1epcx0w4] +c ||[sub:1epcx0w4]s[/sub:1epcx0w4] = sqr root(<ax[sup:1epcx0w4]2[/sup:1epcx0w4] +bx[sup:1epcx0w4]2[/sup:1epcx0w4] +c , ax[sup:1epcx0w4]2[/sup:1epcx0w4] +bx[sup:1epcx0w4]2[/sup:1epcx0w4] +c >)

I know that if a norm is defined from an inner product then <u,v> = 1/2(||u+v||[sup:1epcx0w4]2[/sup:1epcx0w4] - ||u||[sup:1epcx0w4]2[/sup:1epcx0w4] - ||v||[sup:1epcx0w4]2[/sup:1epcx0w4])

Do I start with the above known and try to prove that the sum norm doesn't work for this?

 

[daon]

I would assume it was true, and show that a certain contradiction arises. Since there are certain requirements for an inner product, one should be contradicted taking this route.