Let $V$ be a vector space over $\mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. An inner product on $V$ is a map
\begin{align}
(\cdot,\cdot)_V:V\times V&\to \mathbb{F}
\end{align}
such that for all $u,v,w\in V$ and $a,b\in \mathbb{F}$,
\begin{align}
(au+bv,w)_V&=a(u,w)_V+b(v,w)_V,\\
(u,v)_V&=\overline{(v,u)_V},\\
(v,v)_V&\ge 0,\\
(v,v)_V&=0\iff v=0.
\end{align}