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Inmathematics, in particular infunctional analysis andnonlinear analysis, it is possible to define thederivative of a function between twoFréchet spaces. This notion of differentiation, as it isGateaux derivative between Fréchet spaces, is significantly weaker than thederivative in a Banach space, even between generaltopological vector spaces. Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems fromcalculus hold. In particular, thechain rule is true. With some additional constraints on the Fréchet spaces and functions involved, there is an analog of theinverse function theorem called theNash–Moser inverse function theorem, having wide applications in nonlinear analysis anddifferential geometry.

Formally, the definition of differentiation is identical to theGateaux derivative. Specifically, let${\displaystyle X}$ and${\displaystyle Y}$ be Fréchet spaces,${\displaystyle U\subseteq X}$ be anopen 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$${\displaystyle DF(u)v=\underset{\tau \to 0}{lim}\frac{F(u+v\tau )-F(u)}{\tau }}$$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 acontinuous map.

Higher order derivatives are defined inductively via$${\displaystyle D^{k+1}F(u)\left\{v_1,v_2,\dots ,v_{k+1}\right\}=\underset{\tau \to 0}{lim}\frac{D^{k}F(u+\tau v_{k+1})\left\{v_1,\dots ,v_k\right\}-D^{k}F(u)\left\{v_1,\dots ,v_k\right\}}{\tau }.}$$A function is said to be${\displaystyle C^k}$ if${\displaystyle D^{k}F:U\times X\times X\times \cdots \times X\to Y}$ is continuous. It is${\displaystyle C^{\mathrm{\infty }},}$ orsmooth if it is${\displaystyle C^k}$ for every${\displaystyle k.}$

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$${\displaystyle F(b)-F(a)={\int }_0^{1}DF(a+(b-a)t)\cdot (b-a)dt.}$$
• Thechain 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${\displaystyle DF(u)\left\{x_1,\dots ,x_k\right\}}$ 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$${\displaystyle F(u+h)=F(u)+DF(u)h+\frac{1}{2!}{D}^{2}F(u)\{h,h\}+\cdots +\frac{1}{(k-1)!}{D}^{k-1}F(u)\{h,h,\dots ,h\}+R_k}$$ where the remainder term is given by$${\displaystyle R_k(u,h)=\frac{1}{(k-1)!}{\int }_0^1(1-t{)}^{k-1}{D}^{k}F(u+th)\{h,h,\dots ,h\}dt}$$
• Commutativity of directional derivatives. If${\displaystyle F}$ is${\displaystyle C^k,}$ then$${\displaystyle D^{k}F(u)\left\{h_1,\dots ,h_k\right\}=D^{k}F(u)\left\{h_{\sigma (1)},\dots ,h_{\sigma (k)}\right\}}$$ for everypermutation σ of${\displaystyle \{1,2,\dots ,k\}.}$

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

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

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

Frequently the Fréchet spaces that arise in practical applications of the derivative enjoy an additional property: they aretame. Roughly speaking, a tame Fréchet space is one which is almost aBanach 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 ofdifferential topology. Within this context, many more techniques from calculus hold. In particular, there are versions of the inverse and implicit function theorems.

• Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis
• Infinite-dimensional vector function – Whose values lie in an infinite-dimensional vector space

• 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.

• Banach spaces
• Differential calculus
• Euclidean geometry
• Functions and mappings
• Generalizations of the derivative
• Topological vector spaces

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