[[Calculus of variations MOC]]
# Functional differential

Le $B$ be a [[Banach space]] where $\mathbb{K} = \mathbb{C}$ or $\mathbb{K} = \mathbb{R}$
and $F : B \to \mathbb{K}$ be a [[Functional]].
The **functional differential** $\delta F[\rho; -]$ of $F$ at $\rho$ is a linear functional on $B$ defined by #m/def/anal/fun/var
$$
\begin{align*}
F[\rho + \phi] - F[\rho] = \delta F[\rho;\phi] + \varepsilon \|\phi\|
\end{align*}
$$
where $\varepsilon \to 0$ as $\|\phi\| \to 0$.
The functional differential is in turn used to define the [[Functional derivative]].

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