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I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...

I need some help with the polarization formula for the complex case ...

Garling's statement of the polarization formulae reads as follows:https://www.physicsforums.com/attachments/7914In the above text from Garling we read the following:" ... ... in the complex case we have the polarization formula \(\displaystyle \langle x,y \rangle = \frac{1}{4} \left( \sum_{ j = 0 }^3 i^j \| x + i^j y \|^2 \right)\) ... ... "

Can someone please demonstrate how to prove that \(\displaystyle \langle x,y \rangle = \frac{1}{4} \left( \sum_{ j = 0 }^3 i^j \| x + i^j y \|^2 \right)\) ...?Help will be appreciated ...

Peter

==========================================================================================***NOTE***

It may help readers of the above post to know Garling's notation and approach to inner-product spaces ... ... so I am providing the same ... as follows:

https://www.physicsforums.com/attachments/7915

https://www.physicsforums.com/attachments/7916

https://www.physicsforums.com/attachments/7917

https://www.physicsforums.com/attachments/7918Hope that helps ...

Peter