Differentiable vector-valued functions from Euclidean space: Difference between revisions
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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>\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>\R</math> or the [[complex numbers]] <math>\C.</math>
== Continuously differentiable vector-valued functions ==
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# 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).
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 ''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
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::<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>
If ''f'' is differentiable at a point then it is continuous at that point.{{sfn | Trèves | 2006 | pp=412-419}}
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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>
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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<sup>k</sup> functions with support in a compact subset ===
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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<sup>k</sup> functions ===
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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 ===
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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=
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