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| In [[mathematics]], in particular in [[functional analysis]] and [[nonlinear analysis]], it is possible to define the [[derivative (generalizations)|derivative]] of a function between two [[Fréchet space]]s. This notion of differentiation is significantly weaker than the [[Fréchet derivative|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]].
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| == Mathematical details ==
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| 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
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| :<math>
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| DF(u)v=\lim_{\tau\rightarrow 0}\frac{F(u+v \tau)-F(u)}{\tau}
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| </math>
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| if the limit exists. One says that ''F'' is continuously differentiable, or ''C''<sup>1</sup> if the limit exists for all ''v'' ∈ ''X'' and the mapping
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| :''DF'':''U'' x ''X'' → ''Y''
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| is a [[continuous (topology)|continuous]] map.
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| Higher order derivatives are defined inductively via
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| :<math>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}.</math> | |
| A function is said to be ''C''<sup>k</sup> if ''D''<sup>k</sup>''F'' : ''U'' x ''X'' x ''X''x ... x ''X'' → ''Y'' is continuous. It is ''C''<sup>∞</sup>, or '''smooth''' if it is ''C''<sup>k</sup> for every ''k''.
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| == Properties ==
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| 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 ''C''<sup>1</sup> functions. Then the following properties hold:
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| * ('''Fundamental theorem of calculus'''.)
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| ::If the line segment from ''a'' to ''b'' lies entirely within ''U'', then
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| ::<math> F(b)-F(a) = \int_0^1 DF(a+(b-a)t)\cdot (b-a) dt</math>.
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| * ('''The chain rule'''.)
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| ::''D''(''G'' o ''F'')(''u'')''x'' = ''DG''(''F''(''u''))''DF''(''u'')''x'' for all ''u'' ε ''U'' and ''x'' ε ''X''.
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| * ('''Linearity'''.)
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| ::''DF''(''u'')''x'' is linear in ''x''.{{citation needed|date=March 2013}} More generally, if ''F'' is ''C''<sup>k</sup>, then ''DF''(''u''){''x''<sub>1</sub>,...,''x''<sub>k</sub>} is multilinear in the x's.
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| * ('''Taylor's theorem with remainder.''')
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| ::Suppose that the line segment between ''u'' ε ''U'' and ''u+h'' lies entirely within ''u''. If ''F'' is ''C''<sup>k</sup> then
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| ::<math>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</math>
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| ::where the remainder term is given by
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| ::<math>R_k(u,h)=\frac{1}{(k-1)!}\int_0^1(1-t)^{k-1}D^kF(u+th)\{h,h,\dots,h\}dt</math>
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| * ('''Commutativity of directional derivatives.''') If ''F'' is ''C''<sup>k</sup>, then
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| ::<math>D^kF(u)\{h_1,...,h_k\}=D^kF(u)\{h_{\sigma(1)},\dots,h_{\sigma(k)}\}</math> for every [[permutation]] σ of {1,2,...,k}.
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| 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.
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| ==Consequences in differential geometry==
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| The existence of a chain rule allows for the definition of a [[manifold (mathematics)|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.
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| ==Tame Fréchet spaces==
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| 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.
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| ==References==
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| # {{cite journal|author=Hamilton, R. S.|authorlink=Richard Hamilton (professor)|title=The inverse function theorem of Nash and Moser|url=http://projecteuclid.org/euclid.bams/1183549049|
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| journal=Bull. AMS.|issue=7|year=1982|pages=65–222|doi=10.1090/S0273-0979-1982-15004-2|volume=7|mr=656198}}
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Differentiation in Frechet spaces}}
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| [[Category:Differential calculus]]
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| [[Category:Generalizations of the derivative]]
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| [[Category:Topological vector spaces]]
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