Fréchet derivative

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Not to be confused with Differentiation in Fréchet spaces.

In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used in physics and in particular, in quantum field theory. The Fréchet derivative should be contrasted to the more general Gâteaux derivative.

Intuitively, the Fréchet derivative generalizes the idea of linear approximation from functions of one variable to functions on Banach spaces.

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[edit] Definition

Let V and W be Banach spaces, and U\subset V be an open subset of V. A function f : U → W is called Fréchet differentiable at x \in U if there exists a bounded linear operator A_x:V\to W such that

\lim_{h \to 0} \frac{ \| f(x + h) - f(x) - A_x(h) \|_{W} }{ \|h\|_{V} } = 0.

The limit here is meant in the usual sense of a limit of a function defined on a metric space (see Functions on metric spaces). (using V and R as the two metric spaces, and the above expression as the function of argument h. As a consequence, it must exist for all sequences \langle h_n\rangle_{n=1}^{n=\infty} of non-zero elements of V which converge to the zero vector h_n{\rightarrow}0.) If the limit exists, we write Df(x) = Ax and call it the derivative of f at x. A function f which is Fréchet differentiable for any point of U, and whose derivative Df(x) is continuous in x on U, is said to be C1.

This notion of derivative is a generalization of the ordinary derivative of a function on the real numbers f : R → R since the linear maps from R to R are just multiplication by a real number. In this case, Df(x) is the function t \mapsto tf'(x) .

[edit] Properties

A function differentiable at a point is continuous at that point.

Differentiation is a linear operation in the following sense: if f and g are two maps V → W which are differentiable at x, and r and s are scalars (two real or complex numbers), then rf + sg is differentiable at x with D(rf + sg)(x) = rD(f)(x) + sD(g)(x).

The chain rule is also valid in this context: if f : U → Y is differentiable at x in U, and g : Y → W is differentiable at y = f(x), then the composition g o f is differentiable in x and the derivative is the composition of the derivatives:

D(g \circ f)(x) = D(g)(f(x)) \circ D(f)(x).

[edit] Finite dimensions

The Fréchet derivative in finite-dimensional spaces is the usual derivative. In particular, it is represented in coordinates by the Jacobi matrix.

Suppose that f is a map, f:U⊂Rn → Rm with U an open set. If f is Fréchet differentiable at a point a ∈ U, then its derivative is

Df(a) : \mathbf{R}^n \to \mathbf{R}^m \quad\mbox{with}\quad Df(a)(v) = J_f(a) \, v,

where Jf(a) denotes the Jacobi matrix of f at a.

Furthermore, the partial derivatives of f are given by

\frac{\partial f}{\partial x_i}(a) = Df(a)(e_i) = J_f(a) \, e_i,

where {ei} is the canonical basis of Rn. Since the derivative is a linear function, we have for all vectors h ∈ Rn that the directional derivative of f along h is given by

Df(a)(h) = \sum_{i=1}^{n} h_i \frac{\partial f}{\partial x_i}(a).

If all partial derivatives of f exist and are continuous, then f is Fréchet differentiable. The converse is not true: a function may be Fréchet differentiable and yet fail to have continuous partial derivatives.

[edit] Relation to the Gâteaux derivative

A function f : U ⊂ V → W is called Gâteaux differentiable at x \in U if f has a directional derivative along all directions at x. This means that there exists a function g : V → W such that

g(h)=\lim_{t \to 0} \frac{ f(x + th) - f(x) }{ t }

for any vector h in V. If f is Fréchet differentiable at x, it is also Gâteaux differentiable there, and g is just the linear operator A = Df(x). However, not every Gâteaux differentiable function is Fréchet differentiable. If f is Gâteaux differentiable on an open set U ⊂ X, then f is Fréchet differentiable if its Gâteaux derivative is linear and bounded at each point of U and the Gâteaux derivative is a continuous map U → L(X,Y).

For example, the real-valued function f of two real variables defined by

f(x, y)= \begin{cases} \frac{x^2y}{x^4+y^2} & \mbox{ if } (x, y)\ne (0, 0)\\ 0 & \mbox{ if } (x, y)=(0, 0) \end{cases}

is Gâteaux differentiable at (0, 0), with its derivative being

g(a, b)= \begin{cases} \frac{a^2}{b} & \mbox{ if } b\ne 0 \\ 0 & \mbox { if } b=0 \end{cases}.

The function g is not a linear operator, so this function is not Fréchet differentiable.

In another situation, the function f given by

f(x, y)= \begin{cases} \frac{x^3y}{x^6+y^2} & \mbox{ if } (x, y)\ne (0, 0)\\ 0 & \mbox{ if } (x, y)=(0, 0) \end{cases}

is Gâteaux differentiable at (0, 0), with its derivative there being g(a, b) = 0 for all (a, b), which is a linear operator. However, f is not continuous at (0, 0) (one can see by approaching the origin along the curve (t, t3)) and therefore f cannot be Fréchet differentiable at the origin.

One more example only works in infinite dimensions. Let X be a Banach space, and φ a discontinuous linear functional on X. Let

f(x) = \|x\|\phi(x).\,

Then f(x) is Gâteaux differentiable at x=0 with derivative 0. However, f(x) is not Fréchet differentiable since the limit

\lim_{x\to 0}\phi(x)

does not exist.

[edit] Higher derivatives

If f is a differentiable function at all points in an open subset U of V, it follows that its derivative

D f : U \to L(V, W) \,

is a function from U to the space L(V, W) of all bounded linear operators from V to W. This function may as well have a derivative, the second order derivative of f, which, by the definition of derivative, will be a map

D^2 f : U \to L\big(V, L(V, W)\big).

To make it easier to work with second-order derivatives, the space on the right-hand side is identified with the Banach space L2(V×V, W) of all continuous bilinear maps from V to W. An element φ in L(V, L(V, W)) is thus identified with ψ in L2(V×V, W) such that for all x and y in V

\varphi(x)(y)=\psi(x, y)\,

(intuitively: a function φ linear in x with φ(x) linear in y is the same as a bilinear function ψ in x and y).

One may differentiate

D^2 f : U \to L^2(V\times V, W) \,

again, to obtain the third order derivative, which at each point will be a trilinear map, and so on. The n-th derivative will be a function

D^n f : U \to L^n(V\times V\times \cdots \times V, W),

taking values in the Banach space of continuous multilinear maps in n arguments from V to W. Recursively, a function f is n+1 times differentiable on U if it is n times differentiable on U and for each x in U there exists a continuous multilinear map A of n+1 arguments such that the limit

\lim_{h_{n+1} \to 0} \frac{ \| D^nf(x + h_{n+1})(h_1, h_2, \dots, h_n) - D^nf(x)(h_1, h_2, \dots, h_n) - A(h_1, h_2, \dots, h_n, h_{n+1}) \| }{ \|h_{n+1}\| } = 0

exists uniformly for h1, h2, ..., hn in bounded sets in V. In that case, A is the n+1st derivative of f at x.

[edit] See also

[edit] References

  • Eric W Weisstein, Emma Previato, Dictionary of Applied Math for Engineers and Scientists, CRC Press, 2002. ISBN 1584880538.
  • James R. Munkres, Analysis on Manifolds, ADDISON-WESLEY Publishing Company, 1990. ISBN 0-201-51035-9.
  • http://www.probability.net. This webpage is mostly about basic probability and measure theory, but there is nice chapter about Frechet derivative in Banach spaces (chapter about Jacobian formula). All the results are given with proof.
  • H. Cartan, Calcul Differentiel, Hermann, Paris, 1967.
  • B. A. Frigyik, S. Srivastava and M. R. Gupta, Introduction to Functional Derivatives, UWEE Tech Report 2008-0001.