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How to prove polarization identity? For any $u, v \in V$ $4 \langle T(u), v \rangle = \langle T(u+v), u+v \rangle - \langle T(u-v), u-v \rangle + i\langle T(u+iv), u+iv \rangle - i\langle T(u-iv), u-iv \rangle$

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What did you try? –  AD. Nov 4 '10 at 13:57
    
What is the property of the $\langle . , . \rangle$ operator? $\langle u, v + w \rangle = ...$ (Hint: in the real case, we speak about bilinear forms). –  Djaian Nov 4 '10 at 13:59
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As a side comment: you've posted 11 questions on this forum so far, most of them are homework type questions with initially incorrectly applied tags (this one also, which I'll fix in a minute). You may get better attention if you at least tag questions in the correct field. You may also garner more good will if you actually accept answers. At least 3 or 4 of the questions you've asked in the past month have received very nice answers which you've ignored. –  Willie Wong Nov 4 '10 at 14:02
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@AD. : I think you can delete the "What" in your question. –  Djaian Nov 4 '10 at 14:03
    
@Djaian: nice joke! :D –  J. M. Nov 4 '10 at 14:06
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1 Answer

Expand RHS, using distributive property and the property that $\langle u,cv\rangle=\bar{c}\langle u,v\rangle$. Simplify and cancel many terms to get the LHS.

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