# 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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def ContinuousMap.unitsLift {X : Type u_1} {M : Type u_2} [TopologicalSpace X] [Monoid M] [TopologicalSpace M] [ContinuousMul M] :
C(X, M ˣ) ≃ 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.

Equations

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

Instances For

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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.

Equations

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

Instances For

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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, M ˣ)) (x : X) :
↑(unitsLift f) x = ↑(f x)
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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) :
↑(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) :
↑((addUnitsLift.symm 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) :
↑((unitsLift.symm 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, M ˣ)) (x : X) :
↑(unitsLift f)⁻¹ x = ↑(f x)⁻¹
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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) :
↑(-addUnitsLift 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) :
↑((unitsLift.symm f) x)⁻¹ = ↑f⁻¹ x
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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) :
↑(-(addUnitsLift.symm f) x) = ↑(-f) 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) => ⋯.unit
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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, R ˣ)

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.

Equations

- ContinuousMap.unitsOfForallIsUnit h = { toFun := fun (x : X) => ⋯.unit, continuous_toFun := ⋯ }

Instances For

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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) :
(unitsOfForallIsUnit h) x = ⋯.unit
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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, R ˣ) (fun (f : C(X, R ˣ)) => { toFun := fun (x : X) => ↑(f x), continuous_toFun := ⋯ }) fun (f : C(X, R)) => ∀ (x : X), IsUnit (f x)
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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} {R : Type u_3} [TopologicalSpace X] [NormedDivisionRing R] [CompleteSpace R] (f : C(X, R)) :
IsUnit f ↔ ∀ (x : X), f x ≠ 0
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theorem ContinuousMap.spectrum_eq_preimage_range {X : Type u_1} {R : Type u_3} {𝕜 : Type u_4} [TopologicalSpace X] [NormedField 𝕜] [NormedDivisionRing R] [Algebra 𝕜 R] [CompleteSpace R] (f : C(X, R)) :
spectrum 𝕜 f = ⇑(algebraMap 𝕜 R) ⁻¹' Set.range ⇑f
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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