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 ∈ 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 v ∈ 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^kF(u+\tau v_{k+1})\{v_1,\dots,v_k\}-D^kF(u)\{v_1,\dots,v_k\}}{\tau}.

A function is said to be Ck if DkF : U x X x Xx ... x X → Y is continuous. It is C∞, or smooth if it is Ck 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:

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.
D(G o F)(u)x = DG(F(u))DF(u)x for all u ε U and x ε X.
DF(u)x is linear in x. More generally, if F is Ck, then DF(u){x1,...,xk} is multilinear in the x's.
Suppose that the line segment between u ε U and u+h lies entirely within u. If F is Ck then
F(u+h)=F(u)+DF(u)h+\frac{1}{2!}D^2F(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^kF(u+th)\{h,h,\dots,h\}dt
D^kF(u)\{h_1,...,h_k\}=D^kF(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.

Consequences in differential geometry

The existence of a chain rule allows for the definition of a manifold modeled on a Frèchet space: a Fréchet manifold. Furthermore, the linearity of the derivative implies that there is an analog of the tangent bundle for Fréchet manifolds.

Tame Fréchet spaces

Frequently the Fréchet spaces that arise in practical applications of the derivative enjoy an additional property: they are tame. Roughly speaking, a tame Fréchet space is one which is almost a Banach space. On tame spaces, it is possible to define a preferred class of mappings, known as tame maps. On the category of tame spaces under tame maps, the underlying topology is strong enough to support a fully fledged theory of differential topology. Within this context, many more techniques from calculus hold. In particular, there are versions of the inverse and implicit function theorems.

References

  1. Hamilton, R. S. (1982). "The inverse function theorem of Nash and Moser". Bull. AMS. 7 (7): 65–222. doi:10.1090/S0273-0979-1982-15004-2. MR 656198.