Definition:Increment/Functional

Definition

Let $S$ be a set of mappings.

Let $y, h: \R \to \R$ be real functions.

Let $J \sqbrk y: S \to \R$ be a functional defined on a normed vector space.

Consider the following difference:

$ \Delta J \sqbrk {y; h} = J \sqbrk {y + h} - J \sqbrk y$

Then $\Delta J \sqbrk {y; h}$ is known as the increment of the functional $J$.

Also defined as

For fixed $y$ an increment of the functional $J$ is merely a functional of $h$ and is denoted by $\Delta J \sqbrk h$.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 1.3$: The Variation of a Functional. A Necessary Condition for an Extremum