Differentiation in Fréchet spaces

In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation is significantly weaker than the derivative in a Banach space. Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems from calculus hold. In particular, the chain rule is true. With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the inverse function theorem called the Nash-Moser inverse function theorem, having wide applications in nonlinear analysis and differential geometry.

Mathematical details

Formally, the definition of differentiation is identical to the Gâteaux derivative. Specifically, let X and Y be Fréchet spaces, U ⊂ X be an open set, and F : U → Y be a function. The directional derivative of F in the direction v ε X is defined by

$DF(u)v=\lim _{\tau \rightarrow 0}{\frac {F(u+v\tau )-F(u)}{\tau }}$

if the limit exists. One says that F is continuously differentiable, or C¹ if the limit exists for all $h\in X$ and the mapping

DF:U x X → Y

is a continuous map.

Higher order derivatives are defined inductively via

$D^{k+1}F(u)\{v_{1},v_{2},\dots ,v_{k+1}\}=\lim _{\tau \rightarrow 0}{\frac {D^{k}F(u+\tau v_{k+1})\{v_{1},\dots ,v_{k}\}-D^{k}F(u)\{v_{1},\dots ,v_{k}\}}{\tau }}$.

A function is said to be C^k if DF : U x X x Xx ... x X → Y is continuous. It is C^∞, or smooth if it is C^k for every k.

Properties

Let X, Y, and Z be Fréchet spaces. Suppose that U is an open subset of X, V is an open subset of Y, and F : U → V, G : V → Z are a pair of C¹ functions. Then the following properties hold:

• (Fundamental theorem of calculus.)

If the line segment from a to b lies entirely within U, then

$F(b)-F(a)=\int _{0}^{1}DF(a+(b-a)t)\cdot (b-a)dt$.

• (The chain rule.)

D(G o F)(u)x = DG(F(u))DF(u)x for all u ε U and x ε X.

• (Linearity.)

DF(u)x is linear in x. More generally, if F is C^k, then DF(u){x₁,...,x_k} is multilinear in the x's.

• (Taylor's theorem with remainder.)

Suppose that the line segment between u ε U and u+h lies entirely within u. If F is C^k then

$F(u+h)=F(u)+DF(u)h+{\frac {1}{2!}}D^{2}F(u)\{h,h\}+\dots +{\frac {1}{(k-1)!}}D^{k-1}F(u)\{h,h,\dots ,h\}+R_{k}$

where the remainder term is given by

$R_{k}(u,h)={\frac {1}{(k-1)!}}\int _{0}^{1}(1-t)^{k-1}D^{k}F(u)\{h,h,\dots ,h\}dt$

• (Commutativity of directional derivatives.) If F is C^k, then

$D^{k}F(u)\{h_{1},...,h_{k}\}=D^{k}F(u)\{h_{\sigma (1)},\dots ,h_{\sigma (k)}\}$ for every permutation σ of {1,2,...,k}.

The proofs of many of these properties rely fundamentally on the fact that it is possible to define the Riemann integral of continuous curves in a Fréchet space.

References

1. Hamilton, R. S., "The inverse function theorem of Nash and Moser", Bull. AMS. 7 (1982) 65-222.