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Differentiation in Fréchet spaces

Formally, the definition of differentiation is identical to the Gateaux derivative. Specifically, let ${\displaystyle X}$ and ${\displaystyle Y}$ be Fréchet spaces, ${\displaystyle U\subseteq X}$ be an open set, and ${\displaystyle F:U\to Y}$ be a function. The directional derivative of ${\displaystyle F}$ in the direction ${\displaystyle v\in X}$ is defined by $${\displaystyle DF($$u)v=\lim _{\tau \to 0}{\frac {F(u+v\tau )-F(u)}{\tau }}} if the limit exists. One says that ${\displaystyle F}$ is continuously differentiable, or ${\displaystyle C^{1}}$ if the limit exists for all ${\displaystyle v\in X}$ and the mapping $${\displaystyle DF:U\times X\to Y}$$ is a continuous map.

Higher order derivatives are defined inductively via $${\displaystyle D^{k+1}F($$u)\left\{v_{1},v_{2},\ldots ,v_{k+1}\right\}=\lim _{\tau \to 0}{\frac {D^{k}F(u+\tau v_{k+1})\left\{v_{1},\ldots ,v_{k}\right\}-D^{k}F(u)\left\{v_{1},\ldots ,v_{k}\right\}}{\tau }}.} A function is said to be ${\displaystyle C^{k}}$if${\displaystyle D^{k}F:U\times X\times X\times \cdots \times X\to Y}$ It is ${\displaystyle C^{\infty },}$orsmooth if it is ${\displaystyle C^{k}}$ for every ${\displaystyle k.}$

Properties[edit]

Let ${\displaystyle X,Y,}$ and ${\displaystyle Z}$ be Fréchet spaces. Suppose that ${\displaystyle U}$ is an open subset of ${\displaystyle X,}$ ${\displaystyle V}$ is an open subset of ${\displaystyle Y,}$ and ${\displaystyle F:U\to V,}$ ${\displaystyle G:V\to Z}$ are a pair of ${\displaystyle C^{1}}$ functions. Then the following properties hold:

• Fundamental theorem of calculus. If the line segment from ${\displaystyle a}$to${\displaystyle b}$ lies entirely within ${\displaystyle U,}$ then $${\displaystyle F($$b)-F(a)=\int _{0}^{1}DF(a+(b-a)t)\cdot (b-a)dt.}
• The chain rule. For all ${\displaystyle u\in U}$ and ${\displaystyle x\in X,}$ $${\displaystyle D(G\circ F)($$u)x=DG(F(u))DF(u)x}
• Linearity. ${\displaystyle DF($u)x} is linear in ${\displaystyle x.}$^{[citation needed]} More generally, if ${\displaystyle F}$is${\displaystyle C^{k},}$ then ${\displaystyle DF($u)\left\{x_{1},\ldots ,x_{k}\right\}} ismultilinear in the ${\displaystyle x}$'s.
• Taylor's theorem with remainder. Suppose that the line segment between ${\displaystyle u\in U}$ and ${\displaystyle u+h}$ lies entirely within ${\displaystyle U.}$If${\displaystyle F}$is${\displaystyle C^{k}}$ then $${\displaystyle F(u+h)=F($$u)+DF(u)h+{\frac {1}{2!}}D^{2}F(u)\{h,h\}+\cdots +{\frac {1}{(k-1)!}}D^{k-1}F(u)\{h,h,\ldots ,h\}+R_{k}} where the remainder term is given by $${\displaystyle R_{k}(u,h)={\frac {1}{(k-1)!}}\int _{0}^{1}(1-t)^{k-1}D^{k}F(u+th)\{h,h,\ldots ,h\}dt}$$
• Commutativity of directional derivatives. If ${\displaystyle F}$is${\displaystyle C^{k},}$ then $${\displaystyle D^{k}F($$u)\left\{h_{1},\ldots ,h_{k}\right\}=D^{k}F(u)\left\{h_{\sigma (1)},\ldots ,h_{\sigma (k)}\right\}} for every permutation σof${\displaystyle \{1,2,\ldots ,k\}.}$

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.

Smooth mappings[edit]

Surprisingly, a mapping between open subset of Fréchet spaces is smooth (infinitely often differentiable) if it maps smooth curves to smooth curves; see Convenient analysis. Moreover, smooth curves in spaces of smooth functions are just smooth functions of one variable more.

Consequences in differential geometry[edit]

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[edit]

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.

See also[edit]

• Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysisPages displaying short descriptions of redirect targets
• Infinite-dimensional vector function – function whose values lie in an infinite-dimensional vector spacePages displaying wikidata descriptions as a fallback

References[edit]

• Hamilton, R. S. (1982). "The inverse function theorem of Nash and Moser". Bull. Amer. Math. Soc. 7 (1): 65–222. doi:10.1090/S0273-0979-1982-15004-2. MR 0656198.