\begin{definition}[Definition:Inner Product Space]
Let $\Bbb F$ be a subfield of $\C$. 
Let $V$ be a vector space over $\Bbb F$.
Let $\innerprod \cdot \cdot : V \times V \to \Bbb F$ be an inner product on $V$. 
We say that $\struct {V, \innerprod \cdot \cdot}$ is an '''inner product space'''.
That is, an '''inner product space''' is a vector space together with an associated inner product.
\end{definition}