Inner product

Let $$V$$ be a (complex) vector space. An inner product on $$V$$ is a mapping (Def 4.1.1 )

$$\langle \cdot, \cdot\rangle : V\times V \to \mathbb{C}$$

that satisfies the following three properties:

(i) (linearity in first entry): $$\langle \alpha \mathbf{v}+\beta \mathbf{w}, \mathbf u\rangle = \alpha \langle \mathbf{v,u} \rangle +\beta\langle \mathbf{w,u}\rangle \,\,\,\forall \mathbf{v,w,u}\in V,\,\,\forall \alpha, \beta \in \mathbb{C}$$

(ii) $$ \langle \mathbf{v,w}\rangle = \bar{\langle \mathbf{w,v}\rangle },\,\,\forall\mathbf{v,w}\in V$$

(iii) $$\langle \mathbf{v,v}\rangle \ge 0,\,\,\forall \mathbf{v}\in V$$ and $$\langle v,v\rangle =0\iff\mathbf{v=0}$$

Note that (i) and (ii) imply that $$\langle v,\alpha \mathbf{w}+\beta\mathbf{u}\rangle =\bar{\alpha}\langle v,w\rangle +\bar{\beta}\langle \mathbf{v,u}\rangle \,\,\forall\mathbf{v,w,u}\in V, \alpha, \beta \in \mathbb{C}.$$. This property is called antilinearity.

A vector space equipped with an inner product is called an inner product space.