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Old qual question here:

Suppse $(V,\langle,\rangle)$ is a Hermitian inner product space, and $T:V\to V$ is a complex linear transformation. Show that the following are equivalent:

  1. For all $v\in V$, $\langle Tv,Tv\rangle=\langle v,v\rangle$
  2. For all $v,w\in V$, $\langle Tv,Tw\rangle=\langle v,w\rangle$

Now obviously $2\implies1$ but I'm not sure how to go the other way.

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Hint. Remember the polarization identity: $$\langle v,w\rangle=\frac{1}{4}(\langle v+w,v+w\rangle-\langle v-w,v-w\rangle+i\langle v+iw,v+iw\rangle-i\langle v-iw,v-iw\rangle)$$

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  • $\begingroup$ So then it's as simple as replacing $v$ with $Tv$ and $w$ with $Tw$, using that $T$ is linear (so $Tv+Tw=T(v+w)$, et cetera) and then we have $\langle T(v+w),T(v+w)\rangle=\langle v+w,v+w\rangle$ (and similarly for the other three) so you get what you need? $\endgroup$ – Logan Tatham Dec 9 '16 at 17:35
  • $\begingroup$ Yes, that's exactly it! $\endgroup$ – C. Falcon Dec 9 '16 at 17:40

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