Fréchet derivative between Banach spaces

On this page the scalar field is the field $\mathbb{R}$ of real numbers: $X$, $Y$ and $Z$ always denote real Banach spaces (Banach space) and $U\subseteq X$ always denotes an open subset of $X$, in the metric topology of the norm (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

Let $f:U\to Y$ be a map and let $x\in U$. For a bounded linear operator $T\in \mathcal{B}(X,Y)$ (A bounded linear operator between normed spaces, The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators) write

$r(h):=f(x+h)-f(x)-Th(h\in X,x+h\in U)\mathrm{.}$

Then $f$ is Fréchet differentiable at $x$ when there is $T\in \mathcal{B}(X,Y)$ with

${\lim }_{\begin{array}{c}h\to 0\\ h\mathrm{\ne }0\end{array}}\frac{\mathrm{\parallel }f(x+h)-f(x)-Th\mathrm{\parallel }}{\mathrm{\parallel }h\mathrm{\parallel }}=0,$

the limit being taken over those $h$ with $x+h\in U$. Such an operator $T$ is a Fréchet derivative of $f$ at $x$, written $T=Df(x)$, and $\mathrm{\parallel }T\mathrm{\parallel }$ is its operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum). When $f$ is Fréchet differentiable at every point of $U$, $f$ is differentiable on $U$ and the derivative is the map $Df:U\to \mathcal{B}(X,Y)$, $x\mapsto Df(x)$.

• The limit is a genuine two-sided limit. Since $U$ is open and $x\in U$, there is $\rho >0$ with $x+h\in U$ whenever $\mathrm{\parallel }h\mathrm{\parallel }<\rho$. The condition above is therefore a statement about a punctured neighbourhood of $0$, and no one-sided or directional restriction is imposed on $h$. The quantity displayed is defined for every $h\mathrm{\ne }0$ with $x+h\in U$, and it is $0$ at $h=0$ by convention only, which is why $h\mathrm{\ne }0$ is written out.

• The $\epsilon$-$\delta$ form. The displayed limit says exactly: for every real $\epsilon >0$ there is a real $\delta >0$ such that $\mathrm{\parallel }r(h)\mathrm{\parallel }\le \epsilon \mathrm{\parallel }h\mathrm{\parallel }\text{whenever }\mathrm{\parallel }h\mathrm{\parallel }<\delta \text{ and }x+h\in U\mathrm{.}$ This is the form used in every estimate on this page, and it is the form in which the remainder is written $r(h)=o(\mathrm{\parallel }h\mathrm{\parallel })$.

• The derivative is unique when it exists, so the notation $Df(x)$ is unambiguous; this is proved as the next item on the page, The Fréchet derivative is unique.

• Differentiability at $x$ implies continuity at $x$. Take $\epsilon =1$ in the previous remark and let $\mathrm{\parallel }h\mathrm{\parallel }<\delta$ with $x+h\in U$. Then $\mathrm{\parallel }f(x+h)-f(x)\mathrm{\parallel }\le \mathrm{\parallel }Th\mathrm{\parallel }+\mathrm{\parallel }r(h)\mathrm{\parallel }\le (\mathrm{\parallel }T\mathrm{\parallel }+1)\mathrm{\parallel }h\mathrm{\parallel },$ so every $\eta >0$ is met by $\mathrm{\parallel }f(x+h)-f(x)\mathrm{\parallel }<\eta$ once $\mathrm{\parallel }h\mathrm{\parallel }<\min \{\delta ,\eta \mathrm{/}(\mathrm{\parallel }T\mathrm{\parallel }+1)\}$ (the case $\mathrm{\parallel }T\mathrm{\parallel }+1$ being positive); this is continuity of $f$ at $x$ in the $\epsilon$-$\delta$ form (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form). No separate hypothesis of continuity is ever needed with differentiability.

• Fréchet, not Gâteaux or directional. The derivative $T$ is required to be a bounded linear operator defined on all of $X$, and the remainder is measured against $\mathrm{\parallel }h\mathrm{\parallel }$ uniformly over all directions. The weaker notions that test only $h=tv$ along single lines, or that require merely $\mathrm{\parallel }r(h)\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }h\mathrm{\parallel }\to 0$ along each such line, are not used anywhere on this page: every statement below is about the Fréchet derivative, and the inverse and implicit function theorems in particular are proved for it.

• Graphs and affine maps. A bounded linear $T\in \mathcal{B}(X,Y)$ is differentiable at every $x\in X$ with $DT(x)=T$, since the remainder vanishes identically; and the derivative of a constant map is $0$. In particular an affine map $x\mapsto y_0+T(x-x_0)$ has derivative $T$ everywhere. These instances are used without further comment.