# polarization identity

Theorem [polarization identity] - Let $X$ be an inner product space over $\mathbb{R}$. The following identity holds for every $x,y\in X$:

 $\langle x,y\rangle=\frac{1}{4}(\|x+y\|^{2}-\|x-y\|^{2})$

If $X$ is an inner product space over $\mathbb{C}$ instead, the identity becomes

 $\langle x,y\rangle=\frac{1}{4}(\|x+y\|^{2}-\|x-y\|^{2})+\frac{1}{4}i(\|x+iy\|^% {2}-\|x-iy\|^{2})$

Remark - This result shows that the inner product of $X$ is determined by the norm. Moreover, it can be shown that if a normed space $V$ the parallelogram law, the above formulas define an inner product compatible with the norm of $V$.

Title polarization identity PolarizationIdentity 2013-03-22 17:37:20 2013-03-22 17:37:20 asteroid (17536) asteroid (17536) 4 asteroid (17536) Theorem msc 46C05