\begin{definition}[Definition:Minimum Value of Functional]
Let $S$ be a set of mappings.
Let $y, \hat y \in S: \R \to \R$ be real functions.
Let $J \sqbrk y: S \to \R$ be a functional.
Let $J$ have a (relative) extremum for $y = \hat y$.
Let $J \sqbrk y - J \sqbrk {\hat y} \ge 0$ in the neighbourhood of $y = \hat y$.
{{Disambiguate|Definition:Neighborhood}}
Then this extremum is called the '''minimum''' of the functional $J$.
\end{definition}