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

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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.

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.

This article presents the theory of ''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 (mathematics)|differentiation]] between arbitrary TVSs.

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>.

== Continuously differentiable vector-valued functions ==

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).

:'''Definition'''{{sfn | Trèves | 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>'''{{sfn | Trèves | 2006 | pp=412-419}} 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>.

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

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

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

Having defined what it means for a function ''f'' to be <math>C^k</math> (or ''{{mvar|k''}} times continuously differentiable), say that ''f'' is '''''{{mvar|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>.

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

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

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 giveGive <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 | 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 ''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 | Trèves | 2006 | pp=412-419}}{{sfn | Trèves | 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 | Trèves | 2006 | pp=412-419}}

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 | Trèves | 2006 | pp=446-451}}

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

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 | Trèves | 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>.

=== Space of compactly support C^k functions ===

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 | 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}\left( \Omega; Y \right)</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 ''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'''.

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 | Trèves | 2006 | pp=412-419}} One replace "all compact <math>K \subseteq \Omega</math>" with "all <math>K := \overline{\Omega_i}</math>".

=== Properties ===

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

Given <math>\phi \in C^{k}\left(\Omega \times \Delta \right),</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\left( \phi \right) : \Delta \to C^{k}\left( \Omega \right)</math> be defined by <math>I_k\left( \phi \right)(y) := \phi_y.</math>.

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>I_{\infty}\big\vert_{C_c^{\infty}\left( \Omega \times \Delta \right)} : C_c^{\infty}\left( \Omega \times \Delta \right) \to C_c^{\infty}\left( \Delta; C_c^{\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.

}}

'''{{math theorem|name=Theorem'''{{sfn | Trèves | 2006 | pp=412-419}} |note=|style=|math_statement=
Let ''Y'' be a Hausdorff locally convex space.

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 <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>J_{y^{\prime}}\big\vert_{C_c^{\infty}\left( \Omega; Y \right)} : C_c^{\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).

}}

== Identification as a tensor product ==

Suppose henceforth that ''Y'' 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 ''Y'';

this bilinear map turns this subspace into a tensor product of <math>C^{k}\left( \Omega \right)</math> and ''Y'', 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 ''Y''.{{sfn | Trèves | 2006 | pp=412-419}}

If ''X'' 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 ''X'' 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 | Trèves | 2006 | pp=446-451}}

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

'''Theorem'''{{sfn | Trèves | 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>.{{sfn | Trèves | 2006 | pp=446-451}}

}}

== See also ==

* [[{{annotated link|Fréchet derivative]]}}

* [[{{annotated link|Injective tensor product]]}}

== References ==

{{reflist|group=note}}

{{Reflist}}

{{reflist}}

== Bibliography ==

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

==Bibliography==

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* {{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 }} <!-- {{sfn | Diestel | 2008 | p=}} -->

* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}} <!-- {{sfn | Grothendieck | 1955 | p=}} -->

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* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | {{{year| 1982 }}} | p=}} -->

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{{Functional analysis}}