\begin{definition}[Definition:Maximum 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 $y = \hat y$.
Suppose, $J \sqbrk y - J \sqbrk {\hat y} \le 0$ in the neighbourhood of $y = \hat y$.
{{explain|Make sure the link to Definition:Neighborhood invokes the correct subpage -- if there is no such appropriate subpage, then write it}}
Then this extremum is called the '''maximum''' of the functional $J$.
\end{definition}