Polarization Identity

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Theorem

Real Vector Space

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\R$.

Let $\norm \cdot$ be the inner product norm for $V$.


Then we have:

$4 \innerprod x y = \norm {x + y}^2 - \norm {x - y}^2$

for all $x, y \in V$.


Complex Vector Space

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\C$.

Let $\norm \cdot$ be the inner product norm on $V$.


Then, we have:

$4 \innerprod x y = \norm {x + y}^2 - \norm {x - y}^2 + i \norm {x + i y}^2 - i \norm {x - iy}^2$

for each $x, y \in V$.