# Definition The *inner product* on a real [[vector space]] is a [[Bilinear Form|bilinear]] symmetric map $\cdot:V\times V\rightarrow \mathbb{R}$ on a [[normed vector space]] equipped with a [[euclidean norm]] that takes the form: $\pmb{x} \cdot \pmb{y} = \frac{1}{4} [n(\pmb{x} + \pmb{y})]^2 - \frac{1}{4} [n(\pmb{x} - \pmb{y})]^2$ It has the following properties: 1. It is *[[Bilinear Form|bilinear]]*: $(a \pmb{x} + b \pmb{y}) \cdot \pmb{z} = a (\pmb{x} \cdot \pmb{z}) + b (\pmb{y} \cdot \pmb{z})$ $\pmb{z} \cdot (a \pmb{x} + b \pmb{y})= a (\pmb{z} \cdot \pmb{x}) + b (\pmb{z} \cdot \pmb{y})$ 2. It is *symmetric* $ \pmb{x} \cdot \pmb{y} = \pmb{y} \cdot \pmb{x} $ 3. It is *positive definite* $\pmb{x} \cdot \pmb{x} \geq 0 \quad \text{and} \quad \pmb{x} \cdot \pmb{x} = 0 \iff \pmb{x} = \pmb{0}$ A [[vector space]] equipped with an inner product is called an [[Inner Product Space|inner product space]].