Differentiable vector-valued functions from Euclidean space: Difference between revisions
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In the mathematical discipline of [[functional analysis]], a '''differentiable vector-valued
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.
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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>\
== Continuously differentiable vector-valued functions ==
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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.
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>
▲:<math>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.
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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>\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</math> 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}}
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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 <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>\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>
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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>\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}}
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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
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>
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For any compact subset <math>K \subseteq \Omega,</math> let
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 [[
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 [[
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>
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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
is a surjective isomorphism of TVSs.
Furthermore, its restriction
is an isomorphism of TVSs (where <math>C_c^\infty\left(\Omega \times \Delta\right)</math> has its canonical LF topology).
}}
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Let <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>
Then
is a continuous linear map;
and furthermore, its restriction
is also continuous (where <math>C_c^\infty(\Omega;Y)</math> has the canonical LF topology).
}}
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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}}
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== See also ==
* {{annotated link|Convenient vector space}}
* {{annotated link|Differentiation in Fréchet spaces}}
* {{annotated link|Fréchet derivative}}
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