nLab Fréchet derivative

Idea

The Fréchet derivative is a kind of functional derivative.

There are various ways to set up differentiation in finite dimensions, the most common being the total derivative and the directional derivatives. In infinite dimensions, these become the Fréchet derivative and the Gâteaux derivative respectively.

The definition of the Fréchet derivative of a function is a generalisation of the notion of the total derivative of a function in finite dimensions. In finite dimensions, the total derivative of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ at a point $x \in \mathbb{R}^n$ is defined to be (assuming that it exists) the unique linear operator $D f_x \colon \mathbb{R}^n \to \mathbb{R}$ such that:

$$\lim_{h \to 0} \frac{f(x + h) - f(x) - D f_x h}{{\|h\|}} = 0$$

As all norms in finite dimensions are norm-equivalent, the choice of norm does not matter.

This generalises most easily to normed vector spaces. As it involves limits, it is generally most convenient to work with Banach spaces.

Definition

Definition

Let $E$ and $F$ be Banach spaces with norms ${\|\cdot\|_E}$ and ${\|\cdot\|_F}$ respectively. Let $U \subseteq E$ be an open subset. A (possibly non-linear) function $f \colon U \to F$ is said to be Fréchet differentiable at $x \in U$ if there is a continuous linear operator $A_x \colon E \to F$ such that:

$$\lim_{h \to 0} \frac{\|f(x + h) - f(x) - A_x h\|_F}{\|h\|_E} = 0$$

The operator $A_x$ is called the Fréchet derivative of $f$ at $x$ and is written $D f_x$ or $D f(x)$.

Properties

If the Fréchet derivative exists, it is unique.

Extensions

With the notation of Definition , a function $f U \supseteq E \to F$ is said to be of class $C^1$ on $U$ if it is Fréchet differentiable throughout $U$ and the resulting function $x \mapsto D f_x$ is continuous as a function $U \to L(E,F)$, where $L(E,F)$ is the Banach space of continuous linear operators from $E$ to $F$.

The differentiability of $x \mapsto D f_x$ can then be questioned, and for $n \in \mathbb{N}$, the class $C^n$ is defined by iteration in the obvious way.