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

Revision as of 01:23, 15 February 2021

Revision as of 01:34, 15 February 2021

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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 spacesspace]]s (TVSs). After this task is completed, it is revealed that the topological vector space <math>C_cC^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\left( \Omega \right) \widehat{\otimes}_{\epsilon} Y</math> of the usual [[space of smooth test functions]] <math>C^k\left( \Omega \right)</math> with <math>Y.</math>

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:

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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\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>

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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\left(K; \mathbb{F}\right).</math>

For any two compact subsets <math>K \subseteq L \subseteq \Omega,</math> the inclusion

:<math>\operatorname{In}_{K}^{L} : C^k(K;Y) \to C^k(L;Y)</math>

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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}}

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