Given $|| u + v || = || u - v ||$ show $\langle u , v\rangle = 0$

real-numbers inner-products vector-spaces orthogonality

Yes and by the way, this is called polarization identity in real Hilbert space: $$ (x,y)=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2) $$

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