Definition

An inner product of an $\mathbb{F}$-Vector Space $V$ assigns to vectors $v_{1},v_{2}\in V$ the number $\langle v_{1}, v_{2} \rangle\in\mathbb{F}$ and the assignment satisfies the following rules: 1. Symmetry$$\langle v_{1}, v_{2} \rangle =\overline{\langle v_{2}, v_{1} \rangle}$$for $v_{1},v_{2}\in V$. 2. Linearity$$\langle a_{1}v_{1}+a_{2}v_{2}, v \rangle=a_{1}\langle v_{1}, v \rangle +a_{2}\langle v_{2}, v \rangle$$for $a_{1},a_{2}\in\mathbb{F}$ 3. Positivity $$\langle v, v \rangle \ge 0$$for $v\in V$ 4. Definiteness $$\langle v, v \rangle =0\iff v=0_{v}$$