Let T be a linear operator on an inner product space $V$. Prove that $\lVert T(x)\rVert = \lVert x\rVert$ for all $x$ in $V$ iff $\langle T(x), T(y)\rangle = \langle x,y\rangle$ for all $x,y$ in $V$.
I tried using $\lVert T(x-y)\rVert = \lVert x-y\rVert$ and I got stuck when I got to $\langle T(x),T(y)\rangle + \langle T(y),T(x)\rangle = \langle x,y\rangle + \langle y,x\rangle $
Please give me some idea how to proceed from there or is there any other way to prove this? Thank you.