Fréchet derivative - Knowlpedia

Let $E,F$ be real normed vector spaces, $U\subset E$ open, and $a\in U$ . The Fréchet derivative of $f:U\to F$ at $a$ is a bounded linear map $Df(a):E\to F$ such that

\lim_{h\to0,\ h\ne0}\frac{\|f(a+h)-f(a)-Df(a)h\|_F}{\|h\|_E}=0.

It is the linear approximation with an error negligible compared with the size of the increment. Completeness of the spaces is not required.

Uniqueness

If $L_1,L_2$ both satisfy the definition, use increments $h=tv$ for a fixed vector $v$ . Dividing the difference of their remainders by $|t|$ and letting $t\to0$ gives $(L_1-L_2)v=0$ . Thus the derivative is unique.

Euclidean spaces

For $E=\mathbb R^n,F=\mathbb R^m$ , the derivative is represented by the Jacobian matrix. Fréchet differentiability implies existence of the partial derivatives; their existence alone does not imply Fréchet differentiability. For $f(x,y)=(x^2y,x+y)$ ,

Df(a,b)=\begin{pmatrix}2ab&a^2\\1&1\end{pmatrix}.

A bounded linear function $f(x)=Ax$ has derivative $Df(a)=A$ at every point.