Differentiable vector-valued functions from Euclidean space: Difference between revisions

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{{Short description|Differentiable function in functional analysis}}

In the mathematical discipline of [[functional analysis]], it is possible to generalize the notion of [[derivative (mathematics)|derivative]] to infinite dimensional [[topological vector space]]s (TVSs) in multiple ways.

{{one source|date=January 2025}}

But when the ___domain of TVS-value functions is a subset of finite-dimensional [[Euclidean space]] then the number of generalizations of the derivative is much more limited and derivatives are more well behaved.

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In the mathematical discipline of [[functional analysis]], a '''differentiable vector-valued function from Euclidean space''' is a [[differentiable]] function valued in a [[topological vector space]] (TVS) whose [[Domain of a function|domains]] is a subset of some [[Dimension (vector space)|finite-dimensional]] [[Euclidean space]].

In the mathematical discipline of [[functional analysis]], itIt is possible to generalize the notion of [[derivativeDerivative (mathematics)|derivative]] to infinitefunctions whose ___domain and codomain are subsets of dimensionalarbitrary [[topological vector space]]s (TVSs) in multiple ways.

But when the ___domain of a TVS-valuevalued functionsfunction is a subset of a finite-dimensional [[Euclidean space]] then themany of these notions become [[logically equivalent]] resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is muchalso more limited[[well-behaved]] andcompared derivativesto arethe moregeneral well behavedcase.

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 (mathematics)|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 isomorphism|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 [[Restriction of a function|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>\C.</math>

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>\CComplex.</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]]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 ===

=== 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 [[Gâteaux derivative]]. They are fundamental to the analysis of maps between two arbitrary [[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.

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 [[GâteauxGateaux derivative]]. They are fundamental to the analysis of maps between two arbitrary [[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 function <math>f : I \to X</math> from a non-degenerate interval <math>I \subseteq \R</math> into a [[topological space]] <math>X</math> is called a '''{{em|curve}}''',or a '''{{em|<math>C^0</math> curve}}''', and it is also said to be '''{{em|<math>0</math>-times continuously differentiable}}'''.

A curve <math>f : I \to X</math> valued in a [[topological vector space]] <math>(X, \tau)</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 in <math>(X, \tau)</math> exists:

A curvecontinuous map <math>f : I \to X</math> from a subset <math>I \subseteq \mathbb{R}</math> that is valued in a [[topological vector space]] <math>(X, \tau)</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, \tau)</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{0 \neq 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]] 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>

A curveIt 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 differentiablecontinuous curvefunction <math>f : I \to X</math> from a non-empty and non-degenerate interval <math>I \subseteq \R</math> into a [[topological space]] <math>X</math> is saidcalled to bea '''{{em|continuously differentiablecurve}}''' or a '''{{em|<math>C^10</math>}}''' if its '''{{em|derivativecurve}}''', which is the induced mapin <math>f^{\prime} = f^{(1)} : I \to X,.</math> is continuous.

Using induction on <math>1 < k \in \mathbb{N},</math> a curve <math>f : I \to X</math> isA '''{{em|<math>k</math>-times[[Path continuously differentiable(topology)|path]]}}''' if itsin <math>k-1^{\text{th}}X</math> derivativeis a curve in <math>f^{(k-1)} : I \to X</math> whose ___domain is continuouslycompact differentiable,while inan which'''{{em|[[Arc case(topology)|arc]]}}''' theor '''{{em|<math>k^{\text{thmvar|C}}<sup>0</mathsup>-derivativearc}}''' ofin <math>fX</math>}}''' is thea mappath in <math>f^{(k)}X</math> :=that \left(f^{(k-1)}\right)^{\prime}is :also Ia \to[[topological Xembedding]].</math>

For any <math>k \in \{ 1, 2, \ldots, \infty \},</math> a curve <math>f : I \to X</math> isvalued calledin a '''{{em|topological vector space <math>C^kX</math>-arc}}''' oris 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> andwhere <math>fit :is Icalled \toa X'''{{em|<math>C^k</math>-arc}}''' if it is analso a path (or equivalently, also a <math>C^0</math>-arc) (i.e.in aaddition [[topologicalto being a <math>C^k</math>-embedding]]).

A curve is called '''{{em|smooth}}''' or '''{{em|infinitely differentiable}}''' if it is <math>k</math>-times continuously differentiable for every integer <math>k.</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 (mathematics)|arc]]}}''' or '''{{em|{{mvar|C}}⁰-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> is called a '''{{em|<math>C^k</math>-arc}}''' or a '''{{em|<math>C^k</math>-embedding }}''' if it is a <math>C^k</math> curve, <math>f^{\prime}(t) \neq 0</math> for every <math>t \in I,</math> and <math>f : I \to X</math> is an <math>C^0</math>-arc (i.e. a [[topological 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 valueddefined on open subsets of finite-dimensional [[Euclidean space]]s.

Throughout, 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> and let <math>Y</math> be a [[topological vector space]] (TVS).

Suppose <math>t = \left( t_1, \ldots, t_n \right) \in \Omega</math> and <math>f : \operatorname{Dom} f \to Y</math> is a function such that <math>t \in \operatorname{Dom} f</math> with <math>t</math> a limit point of <math>\operatorname{Dom} 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>}}''', such that

::<math>\lim_{\stackrel{p \to t}{p \in \operatorname{Dom} f}} \frac{f(p) - f(t) - \sum_{i=1}^n \left(p_i - t_i \right) e_i}{\|p - t\|_2} = 0</math> in <math>Y</math>

where <math>p = \left(p_1, \ldots, p_n\right).</math>

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{Dom___domain} f \to Y</math> is a function such that <math>t \in \operatorname{Dom___domain} f</math> with <math>t</math> aan limitaccumulation point of <math>\operatorname{Dom___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{Dom___domain} f}} \frac{f(p) - f(t) - \sum_{i=1}^n \left(p_i - t_i \right) e_i}{\|p - t\|_2} = 0</math> \text{ in <math>} 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.

Say that <math>f</math> is <math>C^0</math> if it is continuous.

If <math>f</math> is differentiable atand everyif pointeach inof someits setpartial derivatives is a continuous function then <math>S \subseteq \Omegaf</math> thenis wesaid sayto thatbe ('''{{em|once}}''' or '''{{em|<math>f1</math> is-time}}''') '''{{em|continuously differentiable}}''' inor '''{{em|<math>SC^1.</math>}}'''.{{sfn|Trèves|2006|pp=412–419}}

IfFor <math>fk \in \N,</math> ishaving differentiabledefined atwhat everyit pointmeans offor itsa ___domainfunction and<math>f</math> ifto eachbe of<math>C^k</math> its(or partial<math>k</math> derivativestimes iscontinuously a continuous function then wedifferentiable), 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>}}'''{{sfn|Trèves|2006|pp=412–419}} if <math>f</math> is continuously differentiable and each of its partial derivatives is <math>C^k.</math>

HavingSay defined what it means for a functionthat <math>f</math> to beis <math>C^k{\infty},</math> (or '''{{mvarem|ksmooth}} times continuously differentiable)''', say that <math>fC^\infty,</math> isor '''{{em|<math>k + 1</math> times continuouslyinfinitely differentiable}}''' orif that '''{{em|<math>f</math> is <math>C^{k+1}</math>}}''' iffor all <math>f</math>k is= continuously0, differentiable1, and each of its partial derivatives is <math>C^k\ldots.</math>

Say that <math>f</math> is <math>C^{\infty},</math>The '''{{em|smooth[[Support (mathematics)|support]]}}''' orof '''{{em|infinitely differentiable}}'''a iffunction <math>f</math> is the [[Closure (topology)|closure]] (taken in its ___domain <math>C^k\operatorname{___domain} f</math>) forof allthe set <math>k\{ =x 0,\in 1,\operatorname{___domain} f : f(x) \ldotsneq 0 \}.</math>

If <math>f : \Omega \to Y</math> is any function then its '''{{em|[[Support (mathematics)|support]]}}''' is the [[Closure (topology)|closure]] (taken in <math>\Omega</math>) of the set <math>\{ x \in \operatorname{Dom} f : f(x) \neq 0 \}.</math>

== Spaces of ''C''^''k'' vector-valued functions ==

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{{See also|Distribution (mathematics)}}

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]]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]] <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>

In this section, the definition of the canonical LF-topology on the [[space of smooth test functions]], and the topologies needed for its definition, is generalized to functions valued in general TVSs.

Throughout, let <math>Y</math> be a Hausdorff [[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> and let <math>Y</math> be a [[topological vector space]] (TVS).

=== Space of ''C''^''k'' 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\left( \Omega; Y \right)</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\left( \Omega; \mathbb{F} \right)</math> and <math>C_c^k\left( \Omega \right)</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>l\ell,</math> and <math>\alpha \in A</math> vary in all possible ways.

:<math>\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>l,</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\left( \Omega; Y \right).</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>l\ell,</math> and <math>\alpha \in A</math> vary in all possible ways.{{sfn|Trèves|2006|pp=412–419}}

:<math>\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>l,</math> and <math>\alpha \in A</math> vary in all possible ways.{{sfn|Trèves|2006|pp=412–419}}

If <math>\Omega</math> is a compact space and <math>Y</math> is a Banach space, then <math>C^0\left( \Omega; Y \right)</math> becomes a Banach space normed by <math>\| f \| := \sup_{\omega \in \Omega} \| f(\omega) \|.</math>{{sfn|Trèves|2006|pp=446–451}}

=== Space of ''C''^''k'' 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> let <math>C^k(K; Y)</math> denote the set of all <math>f</math> in <math>C^k\left( \Omega; Y \right)</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 <math>C^k(K; Y)</math>it the subspace topology induced by <math>C^k\left( \Omega; Y \right).</math>{{sfn|Trèves|2006|pp=412–419}}

If <math>\OmegaK</math> is a compact space and <math>Y</math> is a Banach space, then <math>C^0\left( \OmegaK; Y \right)</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\left( K; \mathbb{F} \right).</math>

NoteLet that<math>C^k(K)</math> fordenote <math>C^k(K;\mathbb{F}).</math>
For any two compact subsets <math>K_1K \subseteq K_2L \subseteq \Omega,</math> the natural inclusion
<math display=block>\operatorname{In}_{K_1K}^{K_2L} : C^k\left( K_1K; Y \right) \to C^k\left( K_2L; Y \right)</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\left( \Omega; Y \right).</math>

=== Space of compactly support ''C''^''k'' 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> be
denote the natural inclusion map and giveendow <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]] induced by these map.

The spaces <math>C^k(K;Y)</math> and maps <math>\operatorname{In}_{K_1}^{K_2}</math> form a [[directDirect 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 natural 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 [[directDirect 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 natural injections <math>\operatorname{In}_{\overline{\Omega_i}}.</math>{{sfn|Trèves|2006|pp=412–419}}

Each natural 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\left( K; Y \right)</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}} OneThe replacestatement remains true if "all compact <math>K \subseteq \Omega</math>" is replaced with "all <math>K := \overline{\Omega}_i</math>".

=== Properties ===

{{mathMath 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

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

Furthermore, the 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 when <math>C_c^\infty\left( \Omega \times \Delta \right)</math> has its canonical LF topology.

is an isomorphism of TVSs (where <math>C_c^\infty\left(\Omega \times \Delta\right)</math> has its canonical LF topology).

}}

{{mathMath theorem|name=Theorem{{sfn|Trèves|2006|pp=412-419}}|note=|style=|math_statement=

ForLet <math>Y</math> be a Hausdorff [[Locally convex topological vector space|locally convex]] [[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>

Let <math>Y</math> be a Hausdorff locally convex space.

Then

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, theits 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 a Hausdorff space.

Given a function <math>f \in C^k\left( \Omega \right)</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>\left( f \otimes y \right)(p) = f(p) y.</math>

This defines a bilinear map <math>\otimes : C^k\left( \Omega \right) \times Y \to C^k\left( \Omega; Y \right)</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\left( \Omega \right)</math> and <math>Y,</math> which we will denote by <math>C^k\left( \Omega \right) \otimes Y.</math>{{sfn|Trèves|2006|pp=412–419}}

Furthermore, if <math>C_c^k\left( \Omega \right) \otimes Y</math> denotes the vector subspace of <math>C^k\left( \Omega \right) \otimes Y</math> consisting of all functions with compact support, then <math>C_c^k\left( \Omega \right) \otimes Y</math> is a tensor product of <math>C_c^k\left( \Omega \right)</math> and <math>Y.</math>{{sfn|Trèves|2006|pp=412–419}}

If <math>X</math> is locally compact then <math>C_c^{0}\left( \Omega \right) \otimes Y</math> is dense in <math>C^0\left( \Omega; X \right)</math> while if <math>X</math> is an open subset of <math>\R^{n}</math> then <math>C_c^{\infty}\left( \Omega \right) \otimes Y</math> is dense in <math>C^k\left( \Omega; X \right).</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\left( \Omega; Y \right)</math> is canonically isomorphic to the [[injective tensor product]] <math>C^k\left( \Omega \right) \widehat{\otimes}_{\epsilon} Y.</math>{{sfn|Trèves|2006|pp=446-451}}

}}

== See also ==

* {{annotated link|Convenient vector space}}

* {{annotated link|Crinkled arc}}

* {{annotated link|Differentiation in Fréchet spaces}}

* {{annotated link|Fréchet derivative}}

* {{annotated link|Gateaux derivative}}

* {{annotated link|Infinite-dimensional vector function}}

* {{annotated link|Injective tensor product}}

Line 151 ⟶ 169:

* {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}} <!-- {{sfn|Wong|1979|p=}} -->

{{Analysis in topological vector spaces}}

{{Topological vector spaces}}

{{Functional analysis}}

{{AnalysisInTopologicalVectorSpaces}}

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{{DEFAULTSORT:Differentiable vector-valued functions from Euclidean space}}

[[Category:Functions and mappings]]

[[Category:Banach spaces]]

[[Category:Differential calculus]]

[[Category:Euclidean geometry]]

[[Category:Functions and mappings]]

[[Category:Generalizations of the derivative]]

[[Category:Topological vector spaces]]