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).

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