Suppose $V$ is an inner product space and $T \in \mathcal{L}(V)$. Suppose $\langle Tv, w \rangle = \langle v, w \rangle$ for all $v, w \in V$. Does this imply that $T = I$?

EDIT: Thanks for the replies. In that case, does $\langle Tv, w \rangle = \langle Sv, w \rangle$ imply that $T = S$? Because:

$ \hspace{0.33 in} \langle Tv, w \rangle = \langle Sv, w \rangle$ for all $v, w \in V$

$ \Longrightarrow \langle Tv-Sv, w \rangle = 0$ for all $v, w \in V$

$ \Longrightarrow Tv-Sv = 0$ for all $v \in V$

$ \Longrightarrow T-S = 0$

$ \Longrightarrow T=S$

And the same should hold for $\langle v, Tw \rangle = \langle v, Sw \rangle$, because of additivity in the right slot as well, right?