I have the following question: "Let $V$ be a Hilbert space and let $T$ be a linear operator on $V$. If $S$ is any linear operator on $V$ that satisfies $\langle Tv,w \rangle = \langle Sv,w \rangle$ for all $v,w \in V$, then $S = T$."
My attempt at the problem went something like this: Notice that the equality $\langle (T-S)v,w \rangle = \langle Tv,w \rangle - \langle Sv,w \rangle = 0$ holds for all $v,w \in V$. If $S \neq T$, then $T-S$ is not the zero linear operator and there exists some $v_0 \in V$ such that $(T-S)v_0 = w_0 \neq 0$. But then $0 < \langle (T-S)v_0,w_0 \rangle$ by the properties of the inner product and this is a contradiction.
I have been informed that this argument is incorrect, but cannot seem to find the flaw, nor a correct proof. Any help would be appreciated.