Definition:Differentiable Functional

Definition
Let $S$ be a set of mappings.

Let $y,h\in S:\R\rightarrow\R$ be real functions.

Let $J\sqbrk{y}$, $\phi\sqbrk{y;h}$ be functionals.

Let $\Delta J\sqbrk{y;h}$ be an increment of the functional $J$ such that:


 * $\displaystyle\Delta J\sqbrk{y;h}=\phi\sqbrk{y;h}+\epsilon\size{h}$

Suppose $\phi\sqbrk{y;h}$ is a linear $h$ and


 * $\displaystyle\lim_{\size{h}\to 0}\epsilon=0$.

Then the functional $ J\sqbrk{y}$ is said to be differentiable.