Density of smooth semigroup vectors

Convolving an orbit with a smooth kernel and evaluating at a positive time produces a smooth vector. Shrinking the kernel and the evaluation time to zero approximates any vector. This is the smooth-vector form of the standard regularization argument for strongly continuous semigroups.

References

Engel--Nagel, One-Parameter Semigroups for Linear Evolution Equations, Lemma II.1.3.

theorem TauCeti.Semigroups.StronglyContinuousSemigroup.dense_smoothVectors {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] (S : StronglyContinuousSemigroup X) : Dense ↑S.smoothVectors

Smooth vectors of a strongly continuous semigroup are dense in the Banach space.