Units of continuous functions #

This file concerns itself with C(X, M)ˣ and C(X, Mˣ) when X is a topological space and M has some monoid structure compatible with its topology.

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theorem ContinuousMap.addUnitsLift.proof_6 {X : Type u_2} {M : Type u_1} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : AddUnits C(X, M)) (x : X) :

(↑(-f) + ↑f) x = 0 x

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theorem ContinuousMap.addUnitsLift.proof_9 {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : AddUnits C(X, M)) :

(fun (f : C(X, AddUnits M)) => { val := ContinuousMap.mk fun (x : X) => ↑(f x), neg := ContinuousMap.mk fun (x : X) => ↑(-f x), val_neg := (_ : ((ContinuousMap.mk fun (x : X) => ↑(f x)) + ContinuousMap.mk fun (x : X) => ↑(-f x)) = 0), neg_val := (_ : ((ContinuousMap.mk fun (x : X) => ↑(-f x)) + ContinuousMap.mk fun (x : X) => ↑(f x)) = 0) }) ((fun (f : AddUnits C(X, M)) => ContinuousMap.mk fun (x : X) => { val := ↑f x, neg := ↑(-f) x, val_neg := (_ : (↑f + ↑(-f)) x = 0 x), neg_val := (_ : (↑(-f) + ↑f) x = 0 x) }) f) = f

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theorem ContinuousMap.addUnitsLift.proof_4 {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : C(X, AddUnits M)) :

((ContinuousMap.mk fun (x : X) => ↑(-f x)) + ContinuousMap.mk fun (x : X) => ↑(f x)) = 0

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def ContinuousMap.addUnitsLift {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] :

C(X, AddUnits M) ≃ AddUnits C(X, M)

Equivalence between continuous maps into the additive units of an additive monoid with continuous addition and the additive units of the additive monoid of continuous maps.

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One or more equations did not get rendered due to their size.

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theorem ContinuousMap.addUnitsLift.proof_7 {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : AddUnits C(X, M)) :

Continuous fun (x : X) => { val := ↑f x, neg := ↑(-f) x, val_neg := (_ : (↑f + ↑(-f)) x = 0 x), neg_val := (_ : (↑(-f) + ↑f) x = 0 x) }

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theorem ContinuousMap.addUnitsLift.proof_2 {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : C(X, AddUnits M)) :

Continuous (AddUnits.val ∘ fun (x : X) => -f x)

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theorem ContinuousMap.addUnitsLift.proof_5 {X : Type u_2} {M : Type u_1} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : AddUnits C(X, M)) (x : X) :

(↑f + ↑(-f)) x = 0 x

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theorem ContinuousMap.addUnitsLift.proof_3 {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : C(X, AddUnits M)) :

((ContinuousMap.mk fun (x : X) => ↑(f x)) + ContinuousMap.mk fun (x : X) => ↑(-f x)) = 0

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theorem ContinuousMap.addUnitsLift.proof_1 {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] (f : C(X, AddUnits M)) :

Continuous (AddUnits.val ∘ fun (x : X) => f x)

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theorem ContinuousMap.addUnitsLift.proof_8 {X : Type u_2} {M : Type u_1} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : C(X, AddUnits M)) :

(fun (f : AddUnits C(X, M)) => ContinuousMap.mk fun (x : X) => { val := ↑f x, neg := ↑(-f) x, val_neg := (_ : (↑f + ↑(-f)) x = 0 x), neg_val := (_ : (↑(-f) + ↑f) x = 0 x) }) ((fun (f : C(X, AddUnits M)) => { val := ContinuousMap.mk fun (x : X) => ↑(f x), neg := ContinuousMap.mk fun (x : X) => ↑(-f x), val_neg := (_ : ((ContinuousMap.mk fun (x : X) => ↑(f x)) + ContinuousMap.mk fun (x : X) => ↑(-f x)) = 0), neg_val := (_ : ((ContinuousMap.mk fun (x : X) => ↑(-f x)) + ContinuousMap.mk fun (x : X) => ↑(f x)) = 0) }) f) = f

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@[simp]

theorem ContinuousMap.val_addUnitsLift_apply_apply {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : C(X, AddUnits M)) (x : X) :

↑(ContinuousMap.addUnitsLift f) x = ↑(f x)

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@[simp]

theorem ContinuousMap.val_addUnitsLift_symm_apply_apply {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : AddUnits C(X, M)) (x : X) :

↑((ContinuousMap.addUnitsLift.symm f) x) = ↑f x

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@[simp]

theorem ContinuousMap.val_unitsLift_apply_apply {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [Monoid M] [TopologicalSpace M] [ContinuousMul M] (f : C(X, Mˣ)) (x : X) :

↑(ContinuousMap.unitsLift f) x = ↑(f x)

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@[simp]

theorem ContinuousMap.val_unitsLift_symm_apply_apply {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [Monoid M] [TopologicalSpace M] [ContinuousMul M] (f : C(X, M)ˣ) (x : X) :

↑((ContinuousMap.unitsLift.symm f) x) = ↑f x

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def ContinuousMap.unitsLift {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [Monoid M] [TopologicalSpace M] [ContinuousMul M] :

C(X, Mˣ) ≃ C(X, M)ˣ

Equivalence between continuous maps into the units of a monoid with continuous multiplication and the units of the monoid of continuous maps.

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One or more equations did not get rendered due to their size.

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@[simp]

theorem ContinuousMap.addUnitsLift_apply_neg_apply {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : C(X, AddUnits M)) (x : X) :

↑(-ContinuousMap.addUnitsLift f) x = ↑(-f x)

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@[simp]

theorem ContinuousMap.unitsLift_apply_inv_apply {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [Monoid M] [TopologicalSpace M] [ContinuousMul M] (f : C(X, Mˣ)) (x : X) :

↑(ContinuousMap.unitsLift f)⁻¹ x = ↑(f x)⁻¹

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@[simp]

theorem ContinuousMap.addUnitsLift_symm_apply_apply_neg' {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : AddUnits C(X, M)) (x : X) :

↑(-(ContinuousMap.addUnitsLift.symm f) x) = ↑(-f) x

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@[simp]

theorem ContinuousMap.unitsLift_symm_apply_apply_inv' {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [Monoid M] [TopologicalSpace M] [ContinuousMul M] (f : C(X, M)ˣ) (x : X) :

↑((ContinuousMap.unitsLift.symm f) x)⁻¹ = ↑f⁻¹ x

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theorem ContinuousMap.continuous_isUnit_unit {X : Type u_1} {R : Type u_3} [TopologicalSpace X] [NormedRing R] [CompleteSpace R] {f : C(X, R)} (h : ∀ (x : X), IsUnit (f x)) :

Continuous fun (x : X) => IsUnit.unit (_ : IsUnit (f x))

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@[simp]

theorem ContinuousMap.unitsOfForallIsUnit_apply {X : Type u_1} {R : Type u_3} [TopologicalSpace X] [NormedRing R] [CompleteSpace R] {f : C(X, R)} (h : ∀ (x : X), IsUnit (f x)) (x : X) :

(ContinuousMap.unitsOfForallIsUnit h) x = IsUnit.unit (_ : IsUnit (f x))

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noncomputable def ContinuousMap.unitsOfForallIsUnit {X : Type u_1} {R : Type u_3} [TopologicalSpace X] [NormedRing R] [CompleteSpace R] {f : C(X, R)} (h : ∀ (x : X), IsUnit (f x)) :

C(X, Rˣ)

Construct a continuous map into the group of units of a normed ring from a function into the normed ring and a proof that every element of the range is a unit.

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ContinuousMap.unitsOfForallIsUnit h = ContinuousMap.mk fun (x : X) => IsUnit.unit (_ : IsUnit (f x))

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instance ContinuousMap.canLift {X : Type u_1} {R : Type u_3} [TopologicalSpace X] [NormedRing R] [CompleteSpace R] :

CanLift C(X, R) C(X, Rˣ) (fun (f : C(X, Rˣ)) => ContinuousMap.mk fun (x : X) => ↑(f x)) fun (f : C(X, R)) => ∀ (x : X), IsUnit (f x)

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One or more equations did not get rendered due to their size.

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theorem ContinuousMap.isUnit_iff_forall_isUnit {X : Type u_1} {R : Type u_3} [TopologicalSpace X] [NormedRing R] [CompleteSpace R] (f : C(X, R)) :

IsUnit f ↔ ∀ (x : X), IsUnit (f x)

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theorem ContinuousMap.isUnit_iff_forall_ne_zero {X : Type u_1} {𝕜 : Type u_4} [TopologicalSpace X] [NormedField 𝕜] [CompleteSpace 𝕜] (f : C(X, 𝕜)) :

IsUnit f ↔ ∀ (x : X), f x ≠ 0

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theorem ContinuousMap.spectrum_eq_range {X : Type u_1} {𝕜 : Type u_4} [TopologicalSpace X] [NormedField 𝕜] [CompleteSpace 𝕜] (f : C(X, 𝕜)) :

spectrum 𝕜 f = Set.range ⇑f

Documentation

Mathlib.Topology.ContinuousFunction.Units

Units of continuous functions #