Suppose $T:V \to V$ is a linear transformation on a complex inner product space $(V,\langle,\rangle)$ and suppose that $$\| T(v) \|=\| v \|$$ for all $v \in V$. Here the norm is the one induced by the inner product. How does this imply that $\langle T(x),T(y)\rangle=\langle x,y\rangle$ for any $x,y \in V$?
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Hint The Hermitian isometry condition $||T v|| = || v ||$ is equivalent to $\langle T v, T v \rangle = \langle v, v \rangle$, and one can use the Hermitian polarization identity to write a Hermitian inner product $\langle a, b \rangle$ as a certain $\Bbb C$-linear combination of expressions of the form $\langle c, c \rangle$.
Travis Willse
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