nLab
Fréchet derivative

Contents

Context

Functional analysis

Variational calculus

Manifolds and cobordisms

Contents

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: nf \colon \mathbb{R}^n \to \mathbb{R} at a point x nx \in \mathbb{R}^n is defined to be (assuming that it exists) the unique linear operator Df x: nD f_x \colon \mathbb{R}^n \to \mathbb{R} such that:

lim h0f(x+h)f(x)Df xhh=0 \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 EE and FF be Banach spaces with norms E{\|\cdot\|_E} and F{\|\cdot\|_F} respectively. Let UEU \subseteq E be an open subset. A (possibly non-linear) function f:UFf \colon U \to F is said to be Fréchet differentiable at xUx \in U if there is a continuous linear operator A x:EFA_x \colon E \to F such that:

lim h0f(x+h)f(x)A xh Fh F=0 \lim_{h \to 0} \frac{\|f(x + h) - f(x) - A_x h\|_F}{\|h\|_F} = 0

The operator A xA_x is called the Fréchet derivative of ff at xx and is written Df xD f_x or Df(x)D f(x).

Properties

If the Fréchet derivative exists, it is unique.

Extensions

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

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

Last revised on November 29, 2017 at 08:38:45. See the history of this page for a list of all contributions to it.