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 $x, y \in V$.