Documentation

Mathlib.Analysis.NormedSpace.Units

The group of units of a complete normed ring #

This file contains the basic theory for the group of units (invertible elements) of a complete normed ring (Banach algebras being a notable special case).

Main results #

The constructions Units.oneSub, Units.add, and Units.ofNearby state, in varying forms, that perturbations of a unit are units. The latter two are not stated in their optimal form; more precise versions would use the spectral radius.

The first main result is Units.isOpen: the group of units of a complete normed ring is an open subset of the ring.

The function Ring.inverse (defined elsewhere), for a ring R, sends a : R to a⁻¹ if a is a unit and 0 if not. The other major results of this file (notably NormedRing.inverse_add, NormedRing.inverse_add_norm and NormedRing.inverse_add_norm_diff_nth_order) cover the asymptotic properties of Ring.inverse (x + t) as t → 0.

@[simp]
theorem Units.val_oneSub {R : Type u_1} [NormedRing R] [CompleteSpace R] (t : R) (h : t < 1) :
(Units.oneSub t h) = 1 - t
def Units.oneSub {R : Type u_1} [NormedRing R] [CompleteSpace R] (t : R) (h : t < 1) :

In a complete normed ring, a perturbation of 1 by an element t of distance less than 1 from 1 is a unit. Here we construct its Units structure.

Equations
  • Units.oneSub t h = { val := 1 - t, inv := ∑' (n : ), t ^ n, val_inv := , inv_val := }
Instances For
    @[simp]
    theorem Units.val_add {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (t : R) (h : t < x⁻¹⁻¹) :
    (x.add t h) = x + t
    def Units.add {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (t : R) (h : t < x⁻¹⁻¹) :

    In a complete normed ring, a perturbation of a unit x by an element t of distance less than ‖x⁻¹‖⁻¹ from x is a unit. Here we construct its Units structure.

    Equations
    Instances For
      @[simp]
      theorem Units.val_ofNearby {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (y : R) (h : y - x < x⁻¹⁻¹) :
      (x.ofNearby y h) = y
      def Units.ofNearby {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (y : R) (h : y - x < x⁻¹⁻¹) :

      In a complete normed ring, an element y of distance less than ‖x⁻¹‖⁻¹ from x is a unit. Here we construct its Units structure.

      Equations
      • x.ofNearby y h = (x.add (y - x) h).copy y (x.add (y - x) h)⁻¹
      Instances For
        theorem Units.isOpen {R : Type u_1} [NormedRing R] [CompleteSpace R] :
        IsOpen {x : R | IsUnit x}

        The group of units of a complete normed ring is an open subset of the ring.

        theorem Units.nhds {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) :
        {x : R | IsUnit x} nhds x

        The nonunits in a complete normed ring are contained in the complement of the ball of radius 1 centered at 1 : R.

        theorem NormedRing.inverse_one_sub {R : Type u_1} [NormedRing R] [CompleteSpace R] (t : R) (h : t < 1) :
        theorem NormedRing.inverse_add {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) :
        ∀ᶠ (t : R) in nhds 0, Ring.inverse (x + t) = Ring.inverse (1 + x⁻¹ * t) * x⁻¹

        The formula Ring.inverse (x + t) = Ring.inverse (1 + x⁻¹ * t) * x⁻¹ holds for t sufficiently small.

        theorem NormedRing.inverse_one_sub_nth_order' {R : Type u_1} [NormedRing R] [CompleteSpace R] (n : ) {t : R} (ht : t < 1) :
        Ring.inverse (1 - t) = iFinset.range n, t ^ i + t ^ n * Ring.inverse (1 - t)
        theorem NormedRing.inverse_one_sub_nth_order {R : Type u_1} [NormedRing R] [CompleteSpace R] (n : ) :
        ∀ᶠ (t : R) in nhds 0, Ring.inverse (1 - t) = iFinset.range n, t ^ i + t ^ n * Ring.inverse (1 - t)
        theorem NormedRing.inverse_add_nth_order {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (n : ) :
        ∀ᶠ (t : R) in nhds 0, Ring.inverse (x + t) = (iFinset.range n, (-x⁻¹ * t) ^ i) * x⁻¹ + (-x⁻¹ * t) ^ n * Ring.inverse (x + t)

        The formula Ring.inverse (x + t) = (∑ i ∈ Finset.range n, (- x⁻¹ * t) ^ i) * x⁻¹ + (- x⁻¹ * t) ^ n * Ring.inverse (x + t) holds for t sufficiently small.

        theorem NormedRing.inverse_one_sub_norm {R : Type u_1} [NormedRing R] [CompleteSpace R] :
        (fun (t : R) => Ring.inverse (1 - t)) =O[nhds 0] fun (_t : R) => 1
        theorem NormedRing.inverse_add_norm {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) :
        (fun (t : R) => Ring.inverse (x + t)) =O[nhds 0] fun (_t : R) => 1

        The function fun t ↦ inverse (x + t) is O(1) as t → 0.

        theorem NormedRing.inverse_add_norm_diff_nth_order {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (n : ) :
        (fun (t : R) => Ring.inverse (x + t) - (iFinset.range n, (-x⁻¹ * t) ^ i) * x⁻¹) =O[nhds 0] fun (t : R) => t ^ n

        The function fun t ↦ Ring.inverse (x + t) - (∑ i ∈ Finset.range n, (- x⁻¹ * t) ^ i) * x⁻¹ is O(t ^ n) as t → 0.

        theorem NormedRing.inverse_add_norm_diff_first_order {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) :
        (fun (t : R) => Ring.inverse (x + t) - x⁻¹) =O[nhds 0] fun (t : R) => t

        The function fun t ↦ Ring.inverse (x + t) - x⁻¹ is O(t) as t → 0.

        theorem NormedRing.inverse_add_norm_diff_second_order {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) :
        (fun (t : R) => Ring.inverse (x + t) - x⁻¹ + x⁻¹ * t * x⁻¹) =O[nhds 0] fun (t : R) => t ^ 2

        The function fun t ↦ Ring.inverse (x + t) - x⁻¹ + x⁻¹ * t * x⁻¹ is O(t ^ 2) as t → 0.

        theorem NormedRing.inverse_continuousAt {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) :
        ContinuousAt Ring.inverse x

        The function Ring.inverse is continuous at each unit of R.

        In a normed ring, the coercion from (equipped with the induced topology from the embedding in R × R) to R is an open embedding.

        theorem Units.isOpenMap_val {R : Type u_1} [NormedRing R] [CompleteSpace R] :
        IsOpenMap Units.val

        In a normed ring, the coercion from (equipped with the induced topology from the embedding in R × R) to R is an open map.

        theorem Ideal.eq_top_of_norm_lt_one {R : Type u_1} [NormedRing R] [CompleteSpace R] (I : Ideal R) {x : R} (hxI : x I) (hx : 1 - x < 1) :
        I =

        An ideal which contains an element within 1 of 1 : R is the unit ideal.

        theorem Ideal.closure_ne_top {R : Type u_1} [NormedRing R] [CompleteSpace R] (I : Ideal R) (hI : I ) :
        I.closure

        The Ideal.closure of a proper ideal in a complete normed ring is proper.

        theorem Ideal.IsMaximal.closure_eq {R : Type u_1} [NormedRing R] [CompleteSpace R] {I : Ideal R} (hI : I.IsMaximal) :
        I.closure = I

        The Ideal.closure of a maximal ideal in a complete normed ring is the ideal itself.

        instance Ideal.IsMaximal.isClosed {R : Type u_1} [NormedRing R] [CompleteSpace R] {I : Ideal R} [hI : I.IsMaximal] :

        Maximal ideals in complete normed rings are closed.

        Equations
        • =