# 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 with respect to $h$ and:

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

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

## Notes

Pragmatically speaking, as $\size h$ approaches $0$, $\epsilon \size h$ can be replaced with sum of terms that are proportional to $\size h^{1 + \delta}$ for $\delta \in \R$ and $\delta > 0$, such that:

- $\displaystyle \lim_{\size h \mathop \to 0} \frac {\size h^{1 + \delta} } {\size h} = \lim_{\size h \mathop \to 0} \size h^\delta = 0$

while at the same time $\phi \sqbrk {y; h}$ becomes proportional to $\size h$, and a similar limit approaches one.

## 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