An inner product space (of vectors over a field \(\mathbb{F}\)) is a vector space, \(V\text{,}\) equipped with the function \((\cdot, \cdot) : V \times V \to \mathbb{F},\) such that, if \(x,y,z \in V\) and \(a \in \mathbb{F}\) then
\begin{equation*} \begin{split} (x,y) \amp =\overline{(x,y)}\\ (ax,y) \amp = a(x,y)\\ (x+y,z) \amp = (x,z)+(y,z)\\ (x,x) \amp \geq 0\\ (x,x) \amp = 0 \text{ iff } x=0. \end{split} \end{equation*} in-context