1. The problem statement, all variables and given/known data Let V be an inner product space. If <x,y> = <x,z> for all x in V, then prove that y = x. 2. Relevant equations 3. The attempt at a solution Since V is a vector space, it follows that (y - z) is an element of V. Since <x,y> = <x,z> for ALL x, we put x = (y-z). Then we have <y - z, y> - <y - z, z> = 0. Since inner products are conjugate linear in the second component, we can write: <y - z, y> - <y - z, z> = <y -z, y - z> = 0. I know that in general <w,w> = 0 if and only if w = 0. Thus we must have y - z = 0, implying that y = z. Is this correct?