# Fréchet derivative

The Fréchet derivative is a derivative defined on Banach spaces, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus variations.

Definition Let $V$ and $W$ be Banach spaces, and $U\subset V$ be an open subset of $V$. A function $f:U\to W$ is called Fréchet differentiable at $x\in U$ if there exists a bounded linear operator $A:V\to W$ such that

$\displaystyle \lim_{h\to 0}\frac{\|f(x+h)-f(x)-Ah\|_W}{\|h\|_V}=0.$

The limit here is meant in the usual sense of a limit of a function defined on  a metric space.

### Relation to the Gâteaux derivative

Definition A function $f:U\subset V\to W$ is called Gâteaux differentiable at $x\in U$ if $f$ has a directional derivative along all direction at $x$. This means that the limit

$\displaystyle \lim_{t\to 0}\frac{f(x+th)-f(x)}{t}$

exists for any choosen vector $h$ in $V$, where is $t$ is from the scalar filed associated with $V$.

If $f$ is Fréchet differentiable at $x$, is is also Gâteaux differentiable there, and the limit is just $Df(x)(h)$.

### Higher derivatives

If $f:U\subset V\to W$ is a differentiable function at all points in an open subset $U$ of $V$, it follows that its derivative

$\displaystyle Df:U\to L(V,W)$

is a function from $U$ to the space $L(V,W)$ of all bounded liner operators from $V$ to $W$. This function may also have a derivative, the second order derivative of $f$, which, by the definition of derivative, will be a map

$\displaystyle D^2f:U\to L(V,L(V,W)).$

To make it easier to work with second order derivatives, the space on the right-hand side is identified with the Banach space $L^2(V\times V,W)$ of all continuous bilinear map from $V$ to $W$. An element $\varphi$ in $L(V,L(V,W))$ is thus identified with $\psi$ in $L^2(V\times V,W)$ such that for all $x$ and $y$ in $V$

$\displaystyle \varphi(x)(y)=\psi(x,y).$