The Fréchet derivative of a function from an open subset of a Banach space into another Banach space at a point is a linear map from the first Banach space into the second one which approximates particularly well the function near the given point. It generalizes the concept of derivative of a real function of one real variable.
If $V$ and $W$ are Banach spaces, $A$ is non-empty subset of $V$, $f\colon A\longrightarrow W$ is a function and $p\in A$, then the Fréchet derivative of $f$ at $p$ is a linear map $A\colon V\longrightarrow W$ such that\begin{equation}\displaystyle\lim_{h\to0}\dfrac{\bigl\|f(p+h)-f(p)-A(h)\bigr\|}{\|h\|}=0.\end{equation}It can be proved that the Fréchet derivative, if it exists, is unique.
This concept generalizes the concept of derivative of a function $f\colon(a,b)\longrightarrow\mathbb R$ at a point $c$, where $a,b,c\in\mathbb R$ are such that $a\lt c\lt b$. Indeed, if this function is diferentiable at $c$ in the usual sense and $f'(c)=m$, then its Fréchet derivative is the linear map from $\mathbb R$ into $\mathbb R$ defined by $x\mapsto mx$.
Fréchet derivatives are named after Maurice Fréchet (1878–1973).
