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Mathlib.Topology.ContinuousMap.Units

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, 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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    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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      @[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)
      source
      @[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)
      source
      @[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
      source
      @[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
      source
      @[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
      source
      @[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)
      source
      @[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
      source
      @[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
      source
      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
      source
      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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        @[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 = ⋯.unit
        source
        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)) => { toFun := fun (x : X) => ↑(f x), continuous_toFun := ⋯ }) fun (f : C(X, R)) => ∀ (x : X), IsUnit (f x)
        source
        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)
        source
        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
        source
        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