1. The problem statement, all variables and given/known data Let V be an inner product space. Then for x,y,z [tex]\in[/tex] V and c[tex]\in[/tex]F, where F is a field denoting either R or C, prove that <x,x> = 0 if and only if x=0. Notes on notation: Here <x,y> denotes the inner product of vectors x and y on some vector space V. 2. Relevant equations <x,y+z> = <x,y> + <x,z> <cx,y> = c<x,y> And a few others. 3. The attempt at a solution It seems pretty straightforward to prove the converse, namely that x=0 implies <0,0> = 0, like this: <-x+x,0> = <-x,0> + <x,0> <0,0> = <0,0> + <0,0> <0,0> = 0. But how do I prove the "forward" conjecture? I know that x=0 iff for some y[tex]\in[/tex] V x+y = y, but I can't start with x, only <x,x> = 0, and I don't see how to "extract" x such that I can show x+y = y. Any thoughts?