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analysis.normed_space.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 one_sub, add and unit_of_nearby 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 is_open: the group of units of a complete normed ring is an open subset of the ring.

The function inverse (defined in algebra.ring), 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 inverse_add, inverse_add_norm and inverse_add_norm_diff_nth_order) cover the asymptotic properties of inverse (x + t) as t → 0.

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@[simp]
theorem units.coe_one_sub {R : Type u_1} [normed_ring R] [complete_space R] (t : R) (h : ∥t∥ < 1) :
↑(units.one_sub t h) = 1 - t
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noncomputable def units.one_sub {R : Type u_1} [normed_ring R] [complete_space R] (t : R) (h : ∥t∥ < 1) :
Rˣ

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.

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noncomputable def units.add {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) (t : R) (h : ∥t∥ < ∥↑x⁻¹∥⁻¹) :
Rˣ

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.

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@[simp]
theorem units.coe_add {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) (t : R) (h : ∥t∥ < ∥↑x⁻¹∥⁻¹) :
↑(x.add t h) = ↑x + t
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noncomputable def units.unit_of_nearby {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) (y : R) (h : ∥y - ↑x∥ < ∥↑x⁻¹∥⁻¹) :
Rˣ

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.

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@[simp]
theorem units.coe_unit_of_nearby {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) (y : R) (h : ∥y - ↑x∥ < ∥↑x⁻¹∥⁻¹) :
↑(x.unit_of_nearby y h) = y
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@[protected]
theorem units.is_open {R : Type u_1} [normed_ring R] [complete_space R] :
is_open {x : R | is_unit x}

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

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@[protected]
theorem units.nhds {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) :
{x : R | is_unit x} ∈ nhds ↑x
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theorem normed_ring.inverse_one_sub {R : Type u_1} [normed_ring R] [complete_space R] (t : R) (h : ∥t∥ < 1) :
ring.inverse (1 - t) = ↑(units.one_sub t h)⁻¹
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theorem normed_ring.inverse_add {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) :
∀ᶠ (t : R) in nhds 0, ring.inverse (↑x + t) = ring.inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹

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

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theorem normed_ring.inverse_one_sub_nth_order {R : Type u_1} [normed_ring R] [complete_space R] (n : ℕ) :
∀ᶠ (t : R) in nhds 0, ring.inverse (1 - t) = (finset.range n).sum (λ (i : ℕ), t ^ i) + t ^ n * ring.inverse (1 - t)
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theorem normed_ring.inverse_add_nth_order {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) (n : ℕ) :
∀ᶠ (t : R) in nhds 0, ring.inverse (↑x + t) = (finset.range n).sum (λ (i : ℕ), (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * ring.inverse (↑x + t)

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

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theorem normed_ring.inverse_one_sub_norm {R : Type u_1} [normed_ring R] [complete_space R] :
(λ (t : R), ring.inverse (1 - t)) =O[nhds 0] λ (t : R), 1
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theorem normed_ring.inverse_add_norm {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) :
(λ (t : R), ring.inverse (↑x + t)) =O[nhds 0] λ (t : R), 1

The function λ t, inverse (x + t) is O(1) as t → 0.

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theorem normed_ring.inverse_add_norm_diff_nth_order {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) (n : ℕ) :
(λ (t : R), ring.inverse (↑x + t) - (finset.range n).sum (λ (i : ℕ), (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[nhds 0] λ (t : R), ∥t∥ ^ n

The function λ t, inverse (x + t) - (∑ i in range n, (- x⁻¹ * t) ^ i) * x⁻¹ is O(t ^ n) as t → 0.

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theorem normed_ring.inverse_add_norm_diff_first_order {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) :
(λ (t : R), ring.inverse (↑x + t) - ↑x⁻¹) =O[nhds 0] λ (t : R), ∥t∥

The function λ t, inverse (x + t) - x⁻¹ is O(t) as t → 0.

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theorem normed_ring.inverse_add_norm_diff_second_order {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) :
(λ (t : R), ring.inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[nhds 0] λ (t : R), ∥t∥ ^ 2

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

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theorem normed_ring.inverse_continuous_at {R : Type u_1} [normed_ring R] [complete_space R] (x : Rˣ) :
continuous_at ring.inverse ↑x

The function inverse is continuous at each unit of R.

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theorem units.is_open_map_coe {R : Type u_1} [normed_ring R] [complete_space R] :
is_open_map coe

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

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theorem units.open_embedding_coe {R : Type u_1} [normed_ring R] [complete_space R] :
open_embedding coe

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