Definition:Differentiable Functional

Definition
Let $S$ be a normed linear space of mappings.

Let $y, h \in S: \R \to \R$ 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:


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

where $\epsilon = \epsilon \sqbrk {y; h}$ is a functional, and $\norm h$ is the norm of $S$.

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


 * $\displaystyle \lim_{\norm h \mathop \to 0} \epsilon = 0$

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