Given two Hilber spaces - $H_1, H_2$ and a transformation $T:H_1 \to H_2$ that is norm preserving and invertable, does this imply that $T$ is also unitary transformation, namely that it preserves the inner product? [or equivalently: is it true that - $a,b, \in H_1 \langle a,b\rangle_{H_1} = \langle T(a),T(b)\rangle_{H_2}$]
Any comments are welcome!
Yes if $T$ is linear, because the norm determines the inner product via the polarization identity $$ \langle x,y\rangle = \frac14 \sum_{n=0}^3 i^n \|x+i^ny\|^2 $$
No if $T$ is allowed to be nonlinear: a counterexample is given by $(r,\theta)\mapsto (r,\theta +\frac12 \sin\theta)$ in polar coordinates in the plane.