Definition:Increment/Functional

Definition
Let $ J \left [ { y } \right ] : S \to \R $ be a functional defined on a normed linear space.

Consider $ J $ evaluated at the "point" $ y + h $, where $ h \left ( { x } \right ) : \R \to \R $ is the increment of an independent variable $ y = y \left ( { x } \right ) $.

Then $ \Delta J \left [ { y; h } \right ] = J \left [ { y + h } \right ] - J \left [ { y } \right ] $ is known as the increment of the functional $ J $.

Also defined as
For fixed y an increment of the functional $ J $  is only a functional of $ h $ and is denoted by $\Delta J[h]$.