View source for Differentiable vector-valued functions from Euclidean space

In the mathematical discipline of [[Functional analysis|functional analysis]], a '''differentiable vector-valued function from Euclidean space''' is a differentiable function valued in a [[Topological vector space|topological vector space]] (TVS) whose [[Domain of a function|domains]] is a subset of some [[Dimension (vector space)|finite-dimensional]] [[Euclidean space]]. 
It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary [[Topological vector space|topological vector space]]s (TVSs) in multiple ways. 
But when the domain of a TVS-valued function is a subset of a finite-dimensional [[Euclidean space]] then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. 
This article presents the theory of <math>k</math>-times continuously differentiable functions on an open subset <math>\Omega</math> of Euclidean space <math>\R^n</math> (<math>1 \leq n < \infty</math>), which is an important special case of differentiation between arbitrary TVSs. 
This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space <math>\R^n</math> so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces. 

All vector spaces will be assumed to be over the field <math>\mathbb{F},</math> where <math>\mathbb{F}</math> is either the real numbers <math>\R</math> or the complex numbers <math>\Complex.</math>

== Continuously differentiable vector-valued functions ==

A map <math>f,</math> which may also be denoted by <math>f^{(0)},</math> between two [[Topological space|topological space]]s is said to be '''{{em|<math>0</math>-times continuously differentiable}}''' or '''{{em|<math>C^0</math>}}''' if it is continuous. A topological embedding may also be called a '''{{em|<math>C^0</math>-embedding}}'''. 

=== Curves ===

Differentiable curves are an important special case of differentiable vector-valued (i.e. TVS-valued) functions which, in particular, are used in the definition of the [[Gateaux derivative]]. They are fundamental to the analysis of maps between two arbitrary [[Topological vector space|topological vector space]]s <math>X \to Y</math> and so also to the analysis of TVS-valued maps from [[Euclidean space]]s, which is the focus of this article. 

A continuous map <math>f : I \to X</math> from a subset <math>I \subseteq \mathbb{R}</math> that is valued in a [[Topological vector space|topological vector space]] <math>X</math> is said to be ('''{{em|once}}''' or '''{{em|<math>1</math>-time}}''') '''{{em|differentiable}}''' if for all <math>t \in I,</math> it is '''{{em|differentiable at <math>t,</math>}}''' which by definition means the following [[Limit of a function#Functions on topological spaces|limit in <math>X</math>]] exists:
<math display=block>f^{\prime}(t) := f^{(1)}(t) 
:= \lim_{\stackrel{r \to t}{t \neq r \in I}} \frac{f(r) - f(t)}{r - t} 
= \lim_{\stackrel{h \to 0}{t \neq t + h \in I}} \frac{f(t + h) - f(t)}{h}</math> 
where in order for this limit to even be well-defined, <math>t</math> must be an [[Accumulation point|accumulation point]] of <math>I.</math> 
If <math>f : I \to X</math> is differentiable then it is said to be '''{{em|continuously differentiable}}''' or '''{{em|<math>C^1</math>}}''' if its '''{{em|derivative}}''', which is the induced map <math>f^{\prime} = f^{(1)} : I \to X,</math> is continuous. 
Using induction on <math>1 < k \in \N,</math> the map <math>f : I \to X</math> is '''{{em|<math>k</math>-times continuously differentiable}}''' or '''{{em|<math>C^k</math>}}''' if its <math>k-1^{\text{th}}</math> derivative <math>f^{(k-1)} : I \to X</math> is continuously differentiable, in which case the '''{{em|<math>k^{\text{th}}</math>-derivative of <math>f</math>}}''' is the map <math>f^{(k)} := \left(f^{(k-1)}\right)^{\prime} : I \to X.</math> 
It is called '''{{em|smooth}}''', <math>C^\infty,</math> or '''{{em|infinitely differentiable}}''' if it is <math>k</math>-times continuously differentiable for every integer <math>k \in \N.</math> 
For <math>k \in \N,</math> it is called '''{{em|<math>k</math>-times differentiable}}''' if it is <math>k-1</math>-times continuous differentiable and <math>f^{(k-1)} : I \to X</math> is differentiable. 

A continuous function <math>f : I \to X</math> from a non-empty and non-degenerate interval <math>I \subseteq \R</math> into a [[Topological space|topological space]] <math>X</math> is called a '''{{em|curve}}''' or a '''{{em|<math>C^0</math> curve}}''' in <math>X.</math> 
A '''{{em|[[Path (topology)|path]]}}''' in <math>X</math> is a curve in <math>X</math> whose domain is compact while an '''{{em|arc}}''' or '''{{em|{{mvar|C}}<sup>0</sup>-arc}}''' in <math>X</math> is a path in <math>X</math> that is also a topological embedding. 
For any <math>k \in \{ 1, 2, \ldots, \infty \},</math> a curve <math>f : I \to X</math> valued in a topological vector space <math>X</math> is called a '''{{em|<math>C^k</math>-embedding }}''' if it is a topological embedding and a <math>C^k</math> curve such that <math>f^{\prime}(t) \neq 0</math> for every <math>t \in I,</math> where it is called a '''{{em|<math>C^k</math>-arc}}''' if it is also a path (or equivalently, also a <math>C^0</math>-arc) in addition to being a <math>C^k</math>-embedding. 

=== Differentiability on Euclidean space ===

The definition given above for curves are now extended from functions valued defined on subsets of <math>\R</math> to functions defined on open subsets of finite-dimensional [[Euclidean space]]s. 

Throughout, let <math>\Omega</math> be an open subset of <math>\R^n,</math> where <math>n \geq 1</math> is an integer. 
Suppose <math>t = \left( t_1, \ldots, t_n \right) \in \Omega</math> and <math>f : \operatorname{domain} f \to Y</math> is a function such that <math>t \in \operatorname{domain} f</math> with <math>t</math> an accumulation point of <math>\operatorname{domain} f.</math>  Then <math>f</math> is '''{{em|differentiable at <math>t</math>}}'''{{sfn|Trèves|2006|pp=412–419}} if there exist <math>n</math> vectors <math>e_1, \ldots, e_n</math> in <math>Y,</math> called the '''{{em|partial derivatives of <math>f</math> at <math>t</math>}}''', such that
<math display=block>\lim_{\stackrel{p \to t}{t \neq p \in \operatorname{domain} f}} \frac{f(p) - f(t) - \sum_{i=1}^n \left(p_i - t_i \right) e_i}{\|p - t\|_2} = 0 \text{ in } Y</math>
where <math>p = \left(p_1, \ldots, p_n\right).</math> 
If <math>f</math> is differentiable at a point then it is continuous at that point.{{sfn|Trèves|2006|pp=412–419}} 
If <math>f</math> is differentiable at every point in some subset <math>S</math> of its domain then <math>f</math> is said to be ('''{{em|once}}''' or '''{{em|<math>1</math>-time}}''') '''{{em|differentiable in <math>S</math>}}''', where if the subset <math>S</math> is not mentioned then this means that it is differentiable at every point in its domain. 
If <math>f</math> is differentiable and if each of its partial derivatives is a continuous function then <math>f</math> is said to be ('''{{em|once}}''' or '''{{em|<math>1</math>-time}}''') '''{{em|continuously differentiable}}''' or '''{{em|<math>C^1.</math>}}'''{{sfn|Trèves|2006|pp=412–419}} 
For <math>k \in \N,</math> having defined what it means for a function <math>f</math> to be <math>C^k</math> (or <math>k</math> times continuously differentiable), say that <math>f</math> is '''{{em|<math>k + 1</math> times continuously differentiable}}''' or that '''{{em|<math>f</math> is <math>C^{k+1}</math>}}''' if <math>f</math> is continuously differentiable and each of its partial derivatives is <math>C^k.</math> 
Say that <math>f</math> is <math>C^{\infty},</math> '''{{em|smooth}}''', <math>C^\infty,</math> or '''{{em|infinitely differentiable}}''' if <math>f</math> is <math>C^k</math> for all <math>k = 0, 1, \ldots.</math> 
The '''{{em|[[Support (mathematics)|support]]}}''' of a function <math>f</math> is the [[Closure (topology)|closure]] (taken in its domain <math>\operatorname{domain} f</math>) of the set <math>\{ x \in \operatorname{domain} f : f(x) \neq 0 \}.</math>

== Spaces of ''C''<sup>''k''</sup> vector-valued functions ==


In this section, the space of smooth test functions and its canonical LF-topology are generalized to functions valued in general [[Complete topological vector space|complete]] Hausdorff locally convex [[Topological vector space|topological vector space]]s (TVSs). After this task is completed, it is revealed that the topological vector space <math>C^k(\Omega;Y)</math> that was constructed could (up to TVS-isomorphism) have instead been defined simply as the completed [[Injective tensor product|injective tensor product]] <math>C^k(\Omega) \widehat{\otimes}_{\epsilon} Y</math> of the usual space of smooth test functions <math>C^k(\Omega)</math> with <math>Y.</math> 

Throughout, let <math>Y</math> be a Hausdorff [[Topological vector space|topological vector space]] (TVS), let <math>k \in \{ 0, 1, \ldots, \infty \},</math> and let <math>\Omega</math> be either:
# an open subset of <math>\R^n,</math> where <math>n \geq 1</math> is an integer, or else
# a locally compact topological space, in which case <math>k</math> can only be <math>0.</math>

=== Space of ''C''<sup>''k''</sup> functions ===

For any <math>k = 0, 1, \ldots, \infty,</math> let <math>C^k(\Omega;Y)</math> denote the vector space of all <math>C^k</math> <math>Y</math>-valued maps defined on <math>\Omega</math> and let <math>C_c^k(\Omega;Y)</math> denote the vector subspace of <math>C^k(\Omega;Y)</math> consisting of all maps in <math>C^k(\Omega;Y)</math> that have compact support. 
Let <math>C^k(\Omega)</math> denote <math>C^k(\Omega;\mathbb{F})</math> and <math>C_c^k(\Omega)</math> denote <math>C_c^k(\Omega; \mathbb{F}).</math> 
Give <math>C_c^k(\Omega;Y)</math> the topology of uniform convergence of the functions together with their derivatives of order <math>< k + 1</math> on the compact subsets of <math>\Omega.</math>{{sfn|Trèves|2006|pp=412–419}}  
Suppose <math>\Omega_1 \subseteq \Omega_2 \subseteq \cdots</math> is a sequence of relatively compact open subsets of <math>\Omega</math> whose union is <math>\Omega</math> and that satisfy <math>\overline{\Omega_i} \subseteq \Omega_{i+1}</math> for all <math>i.</math>
Suppose that <math>\left(V_\alpha\right)_{\alpha \in A}</math> is a basis of neighborhoods of the origin in <math>Y.</math>  Then for any integer <math>\ell < k + 1,</math> the sets:
<math display=block>\mathcal{U}_{i, \ell, \alpha} := \left\{ f \in C^k(\Omega;Y) : \left(\partial / \partial p\right)^q f (p) \in U_\alpha \text{ for all } p \in \Omega_i \text{ and all } q \in \mathbb{N}^n, | q | \leq \ell \right\}</math>
form a basis of neighborhoods of the origin for <math>C^k(\Omega;Y)</math> as <math>i,</math> <math>\ell,</math> and <math>\alpha \in A</math> vary in all possible ways. 
If <math>\Omega</math> is a countable union of compact subsets and <math>Y</math> is a [[Fréchet space]], then so is <math>C^(\Omega;Y).</math> 
Note that <math>\mathcal{U}_{i, l, \alpha}</math> is convex whenever <math>U_{\alpha}</math> is convex. 
If <math>Y</math> is [[Metrizable topological vector space|metrizable]] (resp. [[Complete topological vector space|complete]], [[Locally convex topological vector space|locally convex]], [[Hausdorff space|Hausdorff]]) then so is <math>C^k(\Omega;Y).</math>{{sfn|Trèves|2006|pp=412–419}}{{sfn|Trèves|2006|pp=446–451}} 
If <math>(p_\alpha)_{\alpha \in A}</math> is a basis of continuous seminorms for <math>Y</math> then a basis of continuous seminorms on <math>C^k(\Omega;Y)</math> is:
<math display=block>\mu_{i, l, \alpha}(f) := \sup_{y \in \Omega_i} \left(\sum_{| q | \leq l} p_\alpha\left(\left(\partial / \partial p\right)^q f (p)\right)\right)</math>
as <math>i,</math> <math>\ell,</math> and <math>\alpha \in A</math> vary in all possible ways.{{sfn|Trèves|2006|pp=412–419}}

=== Space of ''C''<sup>''k''</sup> functions with support in a compact subset ===

The definition of the topology of the space of test functions is now duplicated and generalized. 
For any compact subset <math>K \subseteq \Omega,</math> denote the set of all <math>f</math> in <math>C^k(\Omega;Y)</math> whose support lies in <math>K</math> (in particular, if <math>f \in C^k(K;Y)</math> then the domain of <math>f</math> is <math>\Omega</math> rather than <math>K</math>) and give it the subspace topology induced by <math>C^k(\Omega;Y).</math>{{sfn|Trèves|2006|pp=412–419}} 
If <math>K</math> is a compact space and <math>Y</math> is a Banach space, then <math>C^0(K;Y)</math> becomes a Banach space normed by <math>\| f \| := \sup_{\omega \in \Omega} \| f(\omega) \|.</math>{{sfn|Trèves|2006|pp=446–451}} 
Let <math>C^k(K)</math> denote <math>C^k(K;\mathbb{F}).</math> 
For any two compact subsets <math>K \subseteq L \subseteq \Omega,</math> the inclusion 
<math display=block>\operatorname{In}_{K}^{L} : C^k(K;Y) \to C^k(L;Y)</math> 
is an embedding of TVSs and that the union of all <math>C^k(K;Y),</math> as <math>K</math> varies over the compact subsets of <math>\Omega,</math> is <math>C_c^k(\Omega;Y).</math>

=== Space of compactly support ''C''<sup>''k''</sup> functions ===

For any compact subset <math>K \subseteq \Omega,</math> let 
<math display=block>\operatorname{In}_K : C^k(K;Y) \to C_c^k(\Omega;Y)</math> 
denote the inclusion map and endow <math>C_c^k(\Omega;Y)</math> with the strongest topology making all <math>\operatorname{In}_K</math> continuous, which is known as the [[Final topology|final topology]] induced by these map. 
The spaces <math>C^k(K;Y)</math> and maps <math>\operatorname{In}_{K_1}^{K_2}</math> form a [[Direct limit|direct system]] (directed by the compact subsets of <math>\Omega</math>) whose limit in the category of TVSs is <math>C_c^k(\Omega;Y)</math> together with the injections <math>\operatorname{In}_{K}.</math>{{sfn|Trèves|2006|pp=412–419}} 
The spaces <math>C^k\left(\overline{\Omega_i}; Y\right)</math> and maps <math>\operatorname{In}_{\overline{\Omega_i}}^{\overline{\Omega_j}}</math> also form a [[Direct limit|direct system]] (directed by the total order <math>\mathbb{N}</math>) whose limit in the category of TVSs is <math>C_c^k(\Omega;Y)</math> together with the injections <math>\operatorname{In}_{\overline{\Omega_i}}.</math>{{sfn|Trèves|2006|pp=412–419}} 
Each embedding <math>\operatorname{In}_K</math> is an embedding of TVSs. 
A subset <math>S</math> of <math>C_c^k(\Omega;Y)</math> is a neighborhood of the origin in <math>C_c^k(\Omega;Y)</math> if and only if <math>S \cap C^k(K;Y)</math> is a neighborhood of the origin in <math>C^k(K;Y)</math> for every compact <math>K \subseteq \Omega.</math> 
This direct limit topology (i.e. the final topology) on <math>C_c^\infty(\Omega)</math> is known as the '''{{em|canonical LF topology}}'''. 

If <math>Y</math> is a Hausdorff locally convex space, <math>T</math> is a TVS, and <math>u : C_c^k(\Omega;Y) \to T</math> is a linear map, then <math>u</math> is continuous if and only if for all compact <math>K \subseteq \Omega,</math> the restriction of <math>u</math> to <math>C^k(K;Y)</math> is continuous.{{sfn|Trèves|2006|pp=412–419}} The statement remains true if "all compact <math>K \subseteq \Omega</math>" is replaced with "all <math>K := \overline{\Omega}_i</math>". 

=== Properties ===

{{Math theorem|name=Theorem{{sfn|Trèves|2006|pp=412–419}}|note=|style=|math_statement=
Let <math>m</math> be a positive integer and let <math>\Delta</math> be an open subset of <math>\R^m.</math> 
Given <math>\phi \in C^k(\Omega \times \Delta),</math> for any <math>y \in \Delta</math> let <math>\phi_y : \Omega \to \mathbb{F}</math> be defined by <math>\phi_y(x) = \phi(x, y)</math> and let <math>I_k(\phi) : \Delta \to C^k(\Omega)</math> be defined by <math>I_k(\phi)(y) := \phi_y.</math> 
Then 
<math display=block>I_\infty : C^\infty(\Omega \times \Delta) \to C^\infty(\Delta; C^\infty(\Omega))</math> 
is a surjective isomorphism of TVSs. 
Furthermore, its restriction 
<math display=block>I_{\infty}\big\vert_{C_c^{\infty}\left(\Omega \times \Delta\right)} : C_c^\infty(\Omega \times \Delta) \to C_c^\infty\left(\Delta; C_c^\infty(\Omega)\right)</math> 
is an isomorphism of TVSs (where <math>C_c^\infty\left(\Omega \times \Delta\right)</math> has its canonical LF topology). 
}}

{{Math theorem|name=Theorem{{sfn|Trèves|2006|pp=412-419}}|note=|style=|math_statement=
Let <math>Y</math> be a Hausdorff [[Locally convex topological vector space|locally convex]] [[Topological vector space|topological vector space]] and for every continuous linear form <math>y^{\prime} \in Y</math> and every <math>f \in C^\infty(\Omega;Y),</math> let <math>J_{y^{\prime}}(f) : \Omega \to \mathbb{F}</math> be defined by <math>J_{y^{\prime}}(f)(p) = y^{\prime}(f(p)).</math> 
Then 
<math display=block>J_{y^{\prime}} : C^\infty(\Omega;Y) \to C^\infty(\Omega)</math> 
is a continuous linear map; 
and furthermore, its restriction 
<math display=block>J_{y^{\prime}}\big\vert_{C_c^\infty(\Omega;Y)} : C_c^\infty(\Omega;Y) \to C^\infty(\Omega)</math> 
is also continuous (where <math>C_c^\infty(\Omega;Y)</math> has the canonical LF topology).
}}

=== Identification as a tensor product ===

Suppose henceforth that <math>Y</math> is Hausdorff. 
Given a function <math>f \in C^k(\Omega)</math> and a vector <math>y \in Y,</math> let <math>f \otimes y</math> denote the map <math>f \otimes y : \Omega \to Y</math> defined by <math>(f \otimes y)(p) = f(p) y.</math> 
This defines a bilinear map <math>\otimes : C^k(\Omega) \times Y \to C^k(\Omega;Y)</math> into the space of functions whose image is contained in a finite-dimensional vector subspace of <math>Y;</math> 
this bilinear map turns this subspace into a tensor product of <math>C^k(\Omega)</math> and <math>Y,</math> which we will denote by <math>C^k(\Omega) \otimes Y.</math>{{sfn|Trèves|2006|pp=412–419}} 
Furthermore, if <math>C_c^k(\Omega) \otimes Y</math> denotes the vector subspace of <math>C^k(\Omega) \otimes Y</math> consisting of all functions with compact support, then <math>C_c^k(\Omega) \otimes Y</math> is a tensor product of <math>C_c^k(\Omega)</math> and <math>Y.</math>{{sfn|Trèves|2006|pp=412–419}}

If <math>X</math> is locally compact then <math>C_c^{0}(\Omega) \otimes Y</math> is dense in <math>C^0(\Omega;X)</math> while if <math>X</math> is an open subset of <math>\R^{n}</math> then <math>C_c^{\infty}(\Omega) \otimes Y</math> is dense in <math>C^k(\Omega;X).</math>{{sfn|Trèves|2006|pp=446–451}}

{{math theorem|name=Theorem|note=|style=|math_statement=
If <math>Y</math> is a complete Hausdorff locally convex space, then <math>C^k(\Omega;Y)</math> is canonically isomorphic to the [[Injective tensor product|injective tensor product]] <math>C^k(\Omega) \widehat{\otimes}_{\epsilon} Y.</math>{{sfn|Trèves|2006|pp=446-451}}
}}

== See also ==

* {{annotated link|Convenient vector space}}
* {{annotated link|Differentiation in Fréchet spaces}}
* {{annotated link|Fréchet derivative}}
* {{annotated link|Gateaux derivative}}
* {{annotated link|Injective tensor product}}

== Notes ==

{{reflist|group=note}}

== Citations ==

{{reflist}}

== References ==

* {{Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited}} <!-- {{sfn|Diestel|2008|p=}} -->
* {{Dubinsky The Structure of Nuclear Fréchet Spaces}} <!-- {{sfn|Dubinsky|1979|p=}} -->
* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}} <!-- {{sfn|Grothendieck|1955|p=}} -->
* {{Grothendieck Topological Vector Spaces}} <!-- {{sfn|Grothendieck|1973|p=}} -->
* {{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}} <!-- {{sfn|Hogbe-Nlend|Moscatelli|1981|p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} -->
* {{Pietsch Nuclear Locally Convex Spaces|edition=2}} <!-- {{sfn|Pietsch|1979|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Ryan Introduction to Tensor Products of Banach Spaces|edition=1}} <!-- {{sfn|Ryan|2002|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}} <!-- {{sfn|Wong|1979|p=}} -->

{{Functional Analysis}}
{{TopologicalVectorSpaces}}
{{AnalysisInTopologicalVectorSpaces}}

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[[Category:Euclidean geometry]]

{{Sourceattribution|Differentiable vector-valued functions from Euclidean space|1}}