Inner Product

A binary operation $⟨\cdot ,\cdot ⟩:V\times V\to F$ satisfying

Axiom	Description
$⟨v,w⟩=\overline{⟨w,v⟩}$	conjugate symmetry
$⟨v,v⟩\ge 0\forall v\in \mathbb{V}$, with $⟨v,v⟩=0\iff v=0$	positive semidefinitiveness
$⟨v,aw+bz⟩=a⟨v,w⟩+b⟨v,z⟩$	Linearity

where

• $v,w,z\in V$ are vectors in the Vector Space $V$
• $a,b,c\in F$ are scalars in the Field $F$.
• $\overline{c}$ is the complex conjugate of $c\in \mathbb{C}$