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

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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 Hausdorff locally convex topological vector spaces (TVSs). After this task is completed, it is revealed that the topological vector space <math>C_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\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>

In this section, the definition of the canonical LF-topology on the [[space of smooth test functions]], and the topologies needed for its definition, is generalized to functions valued in general TVSs.

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: