Definition:Minimum Value of Functional

Definition
Let $ S $ be a set of mappings.

Let $ y, \hat{ y } \in S $.

Let $ J \left [ { y } \right ] : S \to \R $ be a functional.

Let $ J $ have a (relative) extremum for $ y = \hat { y } $.

Suppose, $ J \left [ { y } \right ] -  J \left [ \hat { y }  \right ] \ge 0 $ in the neighbourhood of $ y = \hat { y }$.

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