Plain-English statement
The observables that possess a right-generator limit form a real vector subspace.
Mathematical statement
D(L) is closed under zero, addition, and real scalar multiplication.
Intuition
Difference quotients are linear in the observable, so their norm limits are linear as well.
Conditions
- a continuous-linear semigroup on a real normed space
- right-generator limits are taken in the ambient norm topology
Why these conditions cannot be dropped
- linearity of each semigroup operator gives linearity of each quotient
- this algebraic domain statement does not imply that the generator is closed
Proof route
- show the zero quotient converges to zero
- add two convergent generator quotients
- scale a convergent quotient by a real scalar
- package the three closure laws as a Submodule
Lean interface notes
- membership retains an existential generator value
- uniqueness is proved separately by tendsto_nhds_unique
Read the mathematics first, then descend into Lean
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Lean statement
def generatorDomainSubmodule (S : ContinuousLinearSemigroup M) :
Submodule ℝ M where
carrier := generatorDomain S
zero_mem' := ⟨0, hasRightGeneratorAt_zero S⟩
add_mem' := by
intro f g hf hg
rcases hf with ⟨Af, hf⟩
rcases hg with ⟨Ag, hg⟩
exact ⟨Af + Ag, hasRightGeneratorAt_add hf hg⟩
smul_mem' := by
intro c f hf
rcases hf with ⟨Af, hf⟩
exact ⟨c • Af, hasRightGeneratorAt_smul hf c⟩
/-- The canonical right-generator value on its submodule domain. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:132published source at 7bcd37294df1