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mathlib_RL_eval_complexity

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import Mathlib.Topology.Algebra.Module.WeakDual import Mathlib.Algebra.Algebra.Spectrum import Mathlib.Topology.ContinuousFunction.Algebra import Mathlib.Data.Set.Lattice #align_import topology.algebra.module.character_space from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64" namespace WeakDual def characterSpace (π•œ : Type*) (A : Type*) [CommSemiring π•œ] [TopologicalSpace π•œ] [ContinuousAdd π•œ] [ContinuousConstSMul π•œ π•œ] [NonUnitalNonAssocSemiring A] [TopologicalSpace A] [Module π•œ A] := {Ο† : WeakDual π•œ A | Ο† β‰  0 ∧ βˆ€ x y : A, Ο† (x * y) = Ο† x * Ο† y} #align weak_dual.character_space WeakDual.characterSpace variable {π•œ : Type*} {A : Type*} -- Porting note: even though the capitalization of the namespace differs, it doesn't matter -- because there is no dot notation since `characterSpace` is only a type via `CoeSort`. namespace CharacterSpace section NonUnitalNonAssocSemiring variable [CommSemiring π•œ] [TopologicalSpace π•œ] [ContinuousAdd π•œ] [ContinuousConstSMul π•œ π•œ] [NonUnitalNonAssocSemiring A] [TopologicalSpace A] [Module π•œ A] instance instFunLike : FunLike (characterSpace π•œ A) A π•œ where coe Ο† := ((Ο† : WeakDual π•œ A) : A β†’ π•œ) coe_injective' Ο† ψ h := by ext1; apply DFunLike.ext; exact congr_fun h instance instContinuousLinearMapClass : ContinuousLinearMapClass (characterSpace π•œ A) π•œ A π•œ where map_smulβ‚›β‚— Ο† := (Ο† : WeakDual π•œ A).map_smul map_add Ο† := (Ο† : WeakDual π•œ A).map_add map_continuous Ο† := (Ο† : WeakDual π•œ A).cont -- Porting note: moved because Lean 4 doesn't see the `DFunLike` instance on `characterSpace π•œ A` -- until the `ContinuousLinearMapClass` instance is declared @[simp, norm_cast] protected theorem coe_coe (Ο† : characterSpace π•œ A) : ⇑(Ο† : WeakDual π•œ A) = (Ο† : A β†’ π•œ) := rfl #align weak_dual.character_space.coe_coe WeakDual.CharacterSpace.coe_coe @[ext] theorem ext {Ο† ψ : characterSpace π•œ A} (h : βˆ€ x, Ο† x = ψ x) : Ο† = ψ := DFunLike.ext _ _ h #align weak_dual.character_space.ext WeakDual.CharacterSpace.ext def toCLM (Ο† : characterSpace π•œ A) : A β†’L[π•œ] π•œ := (Ο† : WeakDual π•œ A) #align weak_dual.character_space.to_clm WeakDual.CharacterSpace.toCLM @[simp] theorem coe_toCLM (Ο† : characterSpace π•œ A) : ⇑(toCLM Ο†) = Ο† := rfl #align weak_dual.character_space.coe_to_clm WeakDual.CharacterSpace.coe_toCLM instance instNonUnitalAlgHomClass : NonUnitalAlgHomClass (characterSpace π•œ A) π•œ A π•œ := { CharacterSpace.instContinuousLinearMapClass with map_smulβ‚›β‚— := fun Ο† => map_smul Ο† map_zero := fun Ο† => map_zero Ο† map_mul := fun Ο† => Ο†.prop.2 } def toNonUnitalAlgHom (Ο† : characterSpace π•œ A) : A →ₙₐ[π•œ] π•œ where toFun := (Ο† : A β†’ π•œ) map_mul' := map_mul Ο† map_smul' := map_smul Ο† map_zero' := map_zero Ο† map_add' := map_add Ο† #align weak_dual.character_space.to_non_unital_alg_hom WeakDual.CharacterSpace.toNonUnitalAlgHom @[simp] theorem coe_toNonUnitalAlgHom (Ο† : characterSpace π•œ A) : ⇑(toNonUnitalAlgHom Ο†) = Ο† := rfl #align weak_dual.character_space.coe_to_non_unital_alg_hom WeakDual.CharacterSpace.coe_toNonUnitalAlgHom instance instIsEmpty [Subsingleton A] : IsEmpty (characterSpace π•œ A) := ⟨fun Ο† => Ο†.prop.1 <| ContinuousLinearMap.ext fun x => by rw [show x = 0 from Subsingleton.elim x 0, map_zero, map_zero] ⟩ variable (π•œ A) theorem union_zero : characterSpace π•œ A βˆͺ {0} = {Ο† : WeakDual π•œ A | βˆ€ x y : A, Ο† (x * y) = Ο† x * Ο† y} := le_antisymm (by rintro Ο† (hΟ† | rfl) Β· exact hΟ†.2 Β· exact fun _ _ => by exact (zero_mul (0 : π•œ)).symm) fun Ο† hΟ† => Or.elim (em <| Ο† = 0) Or.inr fun hβ‚€ => Or.inl ⟨hβ‚€, hΟ†βŸ© #align weak_dual.character_space.union_zero WeakDual.CharacterSpace.union_zero
Mathlib/Topology/Algebra/Module/CharacterSpace.lean
128
134
theorem union_zero_isClosed [T2Space π•œ] [ContinuousMul π•œ] : IsClosed (characterSpace π•œ A βˆͺ {0}) := by
simp only [union_zero, Set.setOf_forall] exact isClosed_iInter fun x => isClosed_iInter fun y => isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _)
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