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

Revision as of 00:42, 15 February 2021

Revision as of 00:47, 15 February 2021

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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:

# an open subset of <math>\R^n,</math> where <math>n \geq 1</math> is an integer, or else

# a [[locally compact]] topological space, in which case <math>k</math> can only be <math>0,.</math> and let <math>Y</math> be a [[topological vector space]] (TVS).

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