View source for Differentiation in Fréchet spaces

{{One source|date=September 2024}}
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, as it is [[Gateaux derivative]] between Fréchet spaces, is significantly weaker than the [[Fréchet derivative|derivative in a Banach space]], even between general [[topological vector space]]s.  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 [[Gateaux derivative]].  Specifically, let <math>X</math> and <math>Y</math> be Fréchet spaces, <math>U \subseteq X</math> be an [[open set]], and <math>F : U \to Y</math> be a function.  The directional derivative of <math>F</math> in the direction <math>v \in X</math> is defined by
<math display="block">DF(u)v = \lim_{\tau \to 0} \frac{F(u + v \tau) - F(u)}{\tau}</math>
if the limit exists.  One says that <math>F</math> is continuously differentiable, or <math>C^1</math> if the limit exists for all <math>v \in X</math> and the mapping
<math display="block">DF : U \times X \to Y</math>
is a [[Continuous (topology)|continuous]] map.

Higher order derivatives are defined inductively via
<math display="block">D^{k+1}F(u) \left\{v_1, v_2, \ldots, v_{k+1}\right\} = \lim_{\tau \to 0} \frac{D^kF(u+\tau v_{k+1}) \left\{v_1, \ldots, v_k\right\} - D^kF(u) \left\{v_1, \ldots, v_k\right\}}{\tau}.</math>
A function is said to be <math>C^k</math> if <math>D^k F : U \times X \times X \times \cdots \times X \to Y</math> is continuous. It is <math>C^{\infty},</math> or '''smooth''' if it is <math>C^k</math> for every <math>k.</math>

== Properties ==

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

* '''[[Fundamental theorem of calculus]]'''. If the line segment from <math>a</math> to <math>b</math> lies entirely within <math>U,</math> then <math display="block">F(b) - F(a) = \int_0^1 DF(a + (b - a) t) \cdot (b - a) dt.</math>
* '''The [[chain rule]]'''. For all <math>u \in U</math> and <math>x \in X,</math> <math display="block">D(G \circ F)(u) x = DG(F(u)) DF(u) x</math> 
* '''[[Linearity]]'''. <math>DF(u) x</math> is linear in <math>x.</math>{{citation needed|date=March 2013}}  More generally, if <math>F</math> is <math>C^k,</math> then <math>DF(u) \left\{ x_1, \ldots, x_k \right\}</math> is [[Multilinear form|multilinear]] in the <math>x</math>'s.
* '''Taylor's theorem with remainder'''. Suppose that the line segment between <math>u \in U</math> and <math>u + h</math> lies entirely within <math>U.</math>  If <math>F</math> is <math>C^k</math> then <math display="block">F(u+h) = F(u) + DF(u)h + \frac{1}{2!} D^2F(u) \{h,h\} + \cdots + \frac{1}{(k-1)!} D^{k-1}F(u) \{h, h, \ldots, h\} + R_k</math> where the remainder term is given by <math display="block">R_k(u,h) = \frac{1}{(k-1)!} \int_0^1(1-t)^{k-1} D^kF(u+th) \{h,h,\ldots,h\}dt</math>
* '''Commutativity of directional derivatives'''. If <math>F</math> is <math>C^k,</math> then <math display="block">D^kF(u) \left\{h_1, \ldots, h_k\right\} = D^kF(u) \left\{h_{\sigma(1)}, \ldots, h_{\sigma(k)}\right\}</math> for every [[permutation]] &sigma; of <math>\{ 1, 2, \ldots, k \}.</math>

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 ==

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 ==

The existence of a chain rule allows for the definition of a [[Manifold (mathematics)|manifold]] modeled on a Fréchet 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 ==
{{Main|Tame Fréchet space}}

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 ==

* {{annotated link|Differentiable vector-valued functions from Euclidean space}}
* {{annotated link|Infinite-dimensional vector function}}

== References ==

{{reflist|group=note}}
{{reflist}}

* {{cite journal|author=Hamilton, R. S.|authorlink=Richard S. Hamilton|title=The inverse function theorem of Nash and Moser|url=http://projecteuclid.org/euclid.bams/1183549049|journal=Bull. Amer. Math. Soc.|issue=1|year=1982|pages=65–222|doi=10.1090/S0273-0979-1982-15004-2|volume=7|mr=656198|doi-access=free}}

{{Functional analysis}}
{{Topological vector spaces}}
{{Analysis in topological vector spaces}}

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[[Category:Banach spaces]]
[[Category:Differential calculus]]
[[Category:Euclidean geometry]]
[[Category:Functions and mappings]]
[[Category:Generalizations of the derivative]]
[[Category:Topological vector spaces]]