# Polarization Identity/Real Vector Space

## Theorem

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 $x, y \in V$.

## Proof

We have:

 $\ds \norm {x + y}^2 - \norm {x - y}^2$ $=$ $\ds \innerprod {x + y} {x + y} - \innerprod {x - y} {x - y}$ Definition of Inner Product Norm $\ds$ $=$ $\ds \paren {\innerprod x {x + y} + \innerprod y {x + y} } - \paren {\innerprod x {x - y} - \innerprod y {x - y} }$ since an inner product is linear in the first argument $\ds$ $=$ $\ds \paren {\innerprod {x + y} x + \innerprod {x + y} y} - \paren {\innerprod {x - y} x - \innerprod {x - y} y}$ since a real inner product is symmetric $\ds$ $=$ $\ds \paren {\innerprod x x + \innerprod y x + \innerprod x y + \innerprod y y} - \paren {\innerprod x x - \innerprod y x - \innerprod x y + \innerprod y y}$ using linearity in the first argument $\ds$ $=$ $\ds 2 \innerprod x y + 2 \innerprod y x$ $\ds$ $=$ $\ds 4 \innerprod x y$ since a real inner product is symmetric

$\blacksquare$