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polarization identity (Theorem)

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

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

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

$\displaystyle \langle x, y \rangle = \frac{1}{4}(\Vert x +y \Vert^2 - \Vert x-y\Vert^2) + \frac{1}{4}i(\Vert x+iy\Vert^2-\Vert x-iy\Vert^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$ satisfies the parallelogram law, the above formulas define an inner product compatible with the norm of $ V$.



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Cross-references: compatible, parallelogram law, normed space, norm, inner product, identity, inner product space
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This is version 1 of polarization identity, born on 2007-11-15.
Object id is 10042, canonical name is PolarizationIdentity3.
Accessed 532 times total.

Classification:
AMS MSC46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )

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