I found in a math textbook this exercise: show that, if A is a linear bounded operator on a complex hilbert space H and <x|Ax> = $0$ for every x in H, then A= $0$. Why the operator is required to be bounded? An hint is specified in the textbook: use the expression <x+cy|A(x+cy)> = $0$, valid for every x,y in H and complex number c. Considering the suggestion:
<x+cy|A(x+cy)>=<x|Ax>+$|c|^2$<y|Ay>+$c^*$<y|Ax>+$c$<x|Ay>= $c^*$<y|Ax>+$c$<x|Ay>=$0$
and, using c=1, <y|Ax>=-<x|Ay>, using c=i, <y|Ax>=<x|Ay> for every x,y in H. So it must be <y|Ax>=$0$ for every x,y in H, which means A=$0$. The problem is I didn't use the boundedness of the operator and, because of that, I'm not sure the proof is correct.
