Definition:Minimum Value of Functional

Definition
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 $y = \hat y$.

Then this extremum is called the minimum of the functional $J$.