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

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

In the field 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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This article presents the theory ''k''-times continuously differentiable functions on an open subset <math>\Omega</math> of Euclidean space <math>\mathbb{R}^n</math> (<math>1 \leq n < \infty</math>), which is an important special case of [[differentiation]] between arbitrary TVSs.

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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>\mathbb{R}</math> or the [[complex numbers]] <math>\mathbb{C}</math>.

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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]].

It is possible to generalize the notion of [[Derivative (mathematics)|derivative]] to functions whose ___domain and codomain are subsets of arbitrary [[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 (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>\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]]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}}'''.

Throughout, let <math>k \in \{ 0, 1, \ldots, \infty \}</math> and let <math>\Omega</math> be either:

# an open subset of <math>\mathbb{R}^n</math>, where <math>n \geq 1</math> is an integer, or else

# a [[locally compact]] topological space, in which ''k'' can only be 0,

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

=== Curves ===

:'''Definition'''{{sfn | Treves | 2006 | pp=412-419}} Suppose <math>p^0 = \left( p^0_1, \ldots, p^0_n \right) \in \Omega</math> and <math>f : \operatorname{Dom} f \to Y</math> is a function such that <math>p^0 \in \operatorname{Dom} f</math> with <math>p^0</math> a limit point of <math>\operatorname{Dom} f</math>. Then we say that ''f'' is '''differentiable at <math>p^0</math>''' if there exist ''n'' vectors <math>e_1, \ldots, e_n</math> in ''Y'', called the '''partial derivatives of ''f''''', such that

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

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

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]]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.

Note that if ''f'' is differentiable at a point then it is continuous at that point.{{sfn | Treves | 2006 | pp=412-419}}

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

If ''f'' is differentiable at every point in some set <math>S \subseteq \Omega</math> then we say that ''f'' is '''differentiable in ''S'''''.

If ''f'' is differentiable at every point of its ___domain and if each of its partial derivatives is a continuous function then we say that ''f'' is '''continuously differentiable''' or <math>C^1</math>.{{sfn | Treves | 2006 | pp=412-419}}

Having defined what it means for a function ''f'' to be <math>C^k</math> (or ''k'' times continuously differentiable), say that ''f'' is '''''k'' + 1 times continuously differentiable''' or that ''f'' is <math>C^{k+1}</math> if ''f'' is continuously differentiable and each of its partial derivatives is <math>C^k</math>.

Say that ''f'' is <math>C^{\infty}</math>, '''smooth''', or '''infinitely differentiable''' if ''f'' is <math>C^{k}</math> for all <math>k = 0, 1, \ldots</math>.

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

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]] <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:

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

<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]] 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]] <math>X</math> is called a '''{{em|curve}}''' or a '''{{em|<math>C^0</math> curve}}''' in <math>X.</math>

=== Space of C^k functions ===

A '''{{em|[[Path (topology)|path]]}}''' in <math>X</math> is a curve in <math>X</math> whose ___domain is compact while an '''{{em|[[Arc (topology)|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> 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 ===

For any <math>k = 0, 1, \ldots, \infty</math>, let <math>C^{k}\left( \Omega; Y \right)</math> denote the vector space of all <math>C^k</math> ''Y''-valued maps defined on <math>\Omega</math> and let <math>C_c^{k}\left( \Omega; Y \right)</math> denote the vector subspace of <math>C^{k}\left( \Omega; Y \right)</math> consisting of all maps in <math>C^{k}\left( \Omega; Y \right)</math> that have compact support.

Let <math>C^{k}\left( \Omega \right)</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}\left( \Omega; \mathbb{F} \right)</math>.

We give <math>C_c^{k}\left( \Omega; Y \right)</math> the topology of uniform convergence of the functions together with their derivatives of order < ''k'' + 1 on the compact subsets of <math>\Omega</math>.{{sfn | Treves | 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 ''i''.

Suppose that <math>\left( V_{\alpha} \right)_{\alpha \in A}</math> is a basis of neighborhoods of the origin in ''Y''.

Then for any integer <math>l < k + 1</math>, the sets:

: <math>\mathcal{U}_{i, l, \alpha} := \left\{ f \in C^{k}\left( \Omega; Y \right) : \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 l \right\}</math>

form a basis of neighborhoods of the origin for <math>C^{k}\left( \Omega; Y \right)</math> as ''i'', ''l'', and <math>\alpha \in A</math> vary in all possible ways.

If <math>\Omega</math> is a countable union of compact subsets and ''Y'' is a [[Fréchet space]], then so is <math>C^{k}\left( \Omega; Y \right)</math>.

Note that <math>\mathcal{U}_{i, l, \alpha}</math> is convex whenever <math>U_{\alpha}</math> is convex.

If ''Y'' is metrizable (resp. complete, locally convex, Hausdorff) then so is <math>C^{k}\left( \Omega; Y \right)</math>.{{sfn | Treves | 2006 | pp=412-419}}{{sfn | Treves | 2006 | pp=446-451}}

If <math>\left( p_{\alpha} \right)_{\alpha \in A}</math> is a basis of continuous seminorms for ''Y'' then a basis of continuous seminorms on <math>C^{k}\left( \Omega; Y \right)</math> is:

:<math>\mu_{i, l, \alpha}\left( f \right) := \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 ''i'', ''l'', and <math>\alpha \in A</math> vary in all possible ways.{{sfn | Treves | 2006 | pp=412-419}}

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.

If <math>\Omega</math> is a compact space and ''Y'' 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 | Treves | 2006 | pp=446-451}}

Throughout, let <math>\Omega</math> be an open subset of <math>\R^n,</math> where <math>n \geq 1</math> is an integer.

=== Space of C^k functions with support in a compact subset ===

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''^''k'' vector-valued functions ==

We now duplicate the definition of the topology of the [[distribution (mathematics)|space of test functions]].

For any compact subset <math>K \subseteq \Omega</math>, let <math>C^{k}\left( K; Y \right)</math> denote the set of all ''f'' in <math>C^{k}\left( \Omega; Y \right)</math> whose support lies in ''K'' (in particular, if <math>f \in C^{k}\left( K; Y \right)</math> then the ___domain of ''f'' is <math>\Omega</math> rather than ''K'') and give <math>C^{k}\left( K; Y \right)</math> the subspace topology induced by <math>C^{k}\left( \Omega; Y \right)</math>.{{sfn | Treves | 2006 | pp=412-419}}

Let <math>C^{k}\left( K \right)</math> denote <math>C^{k}\left( K; \mathbb{F} \right)</math>.

Note that for any two compact subsets <math>K_1 \subseteq K_2 \subseteq \Omega</math>, the natural inclusion <math>\operatorname{In}_{K_1}^{K_2} : C^{k}\left( K_1; Y \right) \to C^{k}\left( K_2; Y \right)</math> is an embedding of TVSs and that the union of all <math>C^{k}\left( K; Y \right)</math>, as ''K'' varies over the compact subsets of <math>\Omega</math>, is <math>C_c^{k}\left( \Omega; Y \right)</math>.

{{See also|Distribution (mathematics)}}

=== Space of compactly support C^k 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]]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>

For any compact subset <math>K \subseteq \Omega</math>, let <math>\operatorname{In}_{K} : C^{k}\left( K; Y \right) \to C_c^{k}\left( \Omega; Y \right)</math> be the natural inclusion and give <math>C_c^{k}\left( \Omega; Y \right)</math> the strongest topology making all <math>\operatorname{In}_{K}</math> continuous.

The spaces <math>C^{k}\left( K; Y \right)</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}\left( \Omega; Y \right)</math> together with the natural injections <math>\operatorname{In}_{K}</math>.{{sfn | Treves | 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}\left( \Omega; Y \right)</math> together with the natural injections <math>\operatorname{In}_{\overline{\Omega_i}}</math>.{{sfn | Treves | 2006 | pp=412-419}}

Each natural embedding <math>\operatorname{In}_{K}</math> is an embedding of TVSs.

A subset ''S'' of <math>C_c^{k}\left( \Omega; Y \right)</math> is a neighborhood of the origin in <math>C_c^{k}\left( \Omega; Y \right)</math> if and only if <math>S \cap C^{k}\left( K; Y \right)</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 on <math>C_c^{\infty}\left( \Omega \right)</math> is known as the '''canonical LF topology'''.

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:

If ''Y'' is a Hausdorff locally convex space, ''T'' is a TVS, and <math>u : C_c^{k}\left( \Omega; Y \right) \to T</math> is a linear map, then ''u'' is continuous if and only if for all compact <math>K \subseteq \Omega</math>, the restriction of ''u'' to <math>C^{k}\left( K; Y \right)</math> is continuous.{{sfn | Treves | 2006 | pp=412-419}} One replace "all compact <math>K \subseteq \Omega</math>" with "all <math>K := \overline{\Omega_i}</math>".

# 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''^''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(\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''^''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> 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''^''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>

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]] 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=

'''Theorem'''{{sfn | Treves | 2006 | pp=412-419}} Let ''m'' be a positive integer and let <math>\Delta</math> be an open subset of <math>\mathbb{R}^{m}</math>.

GivenLet <math>\phi \in C^{k}\left(\Omega \times \Delta \right)m</math>, forbe anya <math>ypositive \ininteger \Delta</math>and let <math>\phi_y : \Omega \to \mathbb{F}Delta</math> be definedan byopen subset of <math>\phi_y(x) = \phi(x, y)R^m.</math>;

andGiven <math>\phi \in C^k(\Omega \times \Delta),</math> for any <math>y \in \Delta</math> let <math>I_k\leftphi_y : \Omega \to \mathbb{F}</math> be defined by <math>\phi_y(x) = \phi(x, y)</math> and let <math>I_k(\rightphi) : \Delta \to C^{k}\left( \Omega \right)</math> be defined by <math>I_k\left( \phi \right)(y) := \phi_y.</math>.

Then

Then <math>I_{\infty} : C^{\infty}\left( \Omega \times \Delta \right) \to C^{\infty}\left( \Delta; C^{\infty}\left( \Omega \right) \right)</math> is a (surjective) isomorphism of TVSs.

Furthermore, the restriction <math display=block>I_{\infty}\big\vert_{C_c^{\infty}\left( \Omega \times \Delta \right)} : C_cC^{\infty}\left( \Omega \times \Delta \right) \to C_cC^{\infty}\left( \Delta; C_cC^{\infty}\left( \Omega \right) \right)</math> is an isomorphism of TVSs when <math>C_c^{\infty}\left( \Omega \times \Delta \right)</math> has its canonical LF topology.

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 | Treves Trèves| 2006 | pp=412-419}} Let ''Y'' be a Hausdorff locally convex space. |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}\left( \Omega; Y \right),</math>, let <math>J_{y^{\prime}}(f) : \Omega \to \mathbb{F}</math> be defined by <math>J_{y^{\prime}}(f)(p) = y^{\prime}\left( f(p) \right).</math>.

Then

Then <math>J_{y^{\prime}} : C^{\infty}\left( \Omega; Y \right) to C^{\infty}\left( \Omega \right)</math> is a continuous linear map;

and furthermore, the restriction <math display=block>J_{y^{\prime}}\big\vert_{C_c^{\infty}\left( \Omega; Y \right)} : C_cC^{\infty}\left( \Omega; Y \right) \to C^{\infty}\left( \Omega \right)</math> is also continuous (where <math>C_c^{\infty}\left( \Omega; Y \right)</math> has the canonical LF topology).

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 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 | Treves Trèves| 2006 | pp=412-419412–419}}

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 | Treves Trèves| 2006 | pp=412-419412–419}}

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>\mathbb{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 | Treves Trèves| 2006 | pp=446-451446–451}}

{{math theorem|name=Theorem|note=|style=|math_statement=

'''Theorem'''{{sfn | Treves | 2006 | pp=446-451}} If ''Y'' 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>.

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]] <math>C^k(\Omega) \widehat{\otimes}_{\epsilon} Y.</math>{{sfn|Trèves|2006|pp=446-451}}

}}

== See also ==

* [[Fréchet derivative]]

* {{annotated link|Convenient vector space}}

* [[Injective tensor product]]

* {{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}}

== Notes ==

{{reflist|group=note}}

== Citations ==

{{reflist}}

== References ==

{{Reflist}}

* {{cite book | last=Diestel | first=Joe | title=The metric theory of tensor products : Grothendieck's résumé revisited | publisher=American Mathematical Society | publication-place=Providence, R.I | year=2008 | isbn=0-8218-4440-7 | oclc=185095773 | ref=harv}} <!-- {{sfn | Diestel | 2008 | p=}} -->

* {{cite book | last=Dubinsky | first=Ed | title=The structure of nuclear Fréchet spaces | publisher=Springer-Verlag | publication-place=Berlin New York | year=1979 | isbn=3-540-09504-7 | oclc=5126156 | ref=harv}} <!-- {{sfn | Dubinsky | 1979 | p=}} -->

* {{cite book | last=Grothendieck | first=Grothendieck | title=Produits tensoriels topologiques et espaces nucléaires | publisher=American Mathematical Society | publication-place=Providence | year=1966 | isbn=0-8218-1216-5 | oclc=1315788 | language=fr | ref=harv}} <!-- {{sfn | Grothendieck | 1966 | p=}} -->

* {{cite book | last=Husain | first=Taqdir | title=Barrelledness in topological and ordered vector spaces | publisher=Springer-Verlag | publication-place=Berlin New York | year=1978 | isbn=3-540-09096-7 | oclc=4493665 | ref=harv}} <!-- {{sfn | Husain | 1978 | p=}} -->

* {{cite book | last=Khaleelulla | first=S. M. | title=Counterexamples in topological vector spaces | publisher=Springer-Verlag | publication-place=Berlin New York | year=1982 | isbn=978-3-540-11565-6 | oclc=8588370 | ref=harv}} <!-- {{sfn | Khaleelulla | 1982 | p=}} -->

* {{cite book | last=Nlend | first=H | title=Bornologies and functional analysis : introductory course on the theory of duality topology-bornology and its use in functional analysis | publisher=North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier-North Holland | publication-place=Amsterdam New York New York | year=1977 | isbn=0-7204-0712-5 | oclc=2798822 | ref=harv}} <!-- {{sfn | Nlend | 1977 | p=}} -->

* {{cite book | last=Nlend | first=H | title=Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality | publisher=North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland | publication-place=Amsterdam New York New York, N.Y | year=1981 | isbn=0-444-86207-2 | oclc=7553061 | ref=harv}} <!-- {{sfn | Nlend | 1981 | p=}} -->

* {{cite book | last=Pietsch | first=Albrecht | title=Nuclear locally convex spaces | publisher=Springer-Verlag | publication-place=Berlin,New York | year=1972 | isbn=0-387-05644-0 | oclc=539541 | ref=harv}} <!-- {{sfn | Pietsch | 1972 | p=}} -->

* {{cite book | last=Robertson | first=A. P. | title=Topological vector spaces | publisher=University Press | publication-place=Cambridge England | year=1973 | isbn=0-521-29882-2 | oclc=589250 | ref=harv}} <!-- {{sfn | Robertson | 1973 | p=}} -->

* {{cite book | last=Ryan | first=Raymond | title=Introduction to tensor products of Banach spaces | publisher=Springer | publication-place=London New York | year=2002 | isbn=1-85233-437-1 | oclc=48092184 | ref=harv}} <!-- {{sfn | Ryan | 2002 | p=}} -->

* {{cite book | last=Schaefer | first=Helmut H.| title=Topological Vector Spaces | publisher=Springer New York Imprint Springer | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | publication-place=New York, NY | year=1999 | isbn=978-1-4612-7155-0 | oclc=840278135 | ref=harv}} <!-- {{sfn | Schaefer | 1999 | p=}} -->

* {{cite book | last=Treves | first=François | title=Topological vector spaces, distributions and kernels | publisher=Dover Publications | publication-place=Mineola, N.Y | year=2006 | isbn=978-0-486-45352-1 | oclc=853623322 | ref=harv}} <!-- {{sfn | Treves | 2006 | p=}} -->

* {{cite book | author=Wong | title=Schwartz spaces, nuclear spaces, and tensor products | publisher=Springer-Verlag | publication-place=Berlin New York | year=1979 | isbn=3-540-09513-6 | oclc=5126158 | ref=harv}} <!-- {{sfn | Wong | 1979 | p=}} -->

* {{Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited}} <!-- {{sfn|Diestel|2008|p=}} -->

{{Template:Functional analysis}}

* {{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=}} -->

{{Analysis in topological vector spaces}}

{{Topological vector spaces}}

{{Functional analysis}}

<!--- Categories --->

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

[[Category:Banach spaces]]

[[Category:Differential calculus]]

[[Category:Euclidean geometry]]

[[Category:Functions and mappings]]

[[Category:Generalizations of the derivative]]

[[Category:Topological vector spaces]]