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refactor(analysis/normed_space/bounded_linear_maps): nondiscrete norm…
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@sgouezel @avigad
sgouezel authored and avigad committed Apr 6, 2019
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245 changes: 108 additions & 137 deletions src/analysis/normed_space/operator_norm.lean
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@@ -1,7 +1,7 @@
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow
Authors: Jan-David Salchow, Sébastien Gouëzel
The space of bounded linear maps
Expand All @@ -20,8 +20,10 @@ variables {k : Type*}
variables {E : Type*} {F : Type*}


-- Define the subspace of bounded linear maps, introduce the notation L(E,F) for the set of bounded linear maps.
-- Define the subspace of bounded linear maps.
section bounded_linear_maps
set_option class.instance_max_depth 50


variables [hnfk : normed_field k] [normed_space k E] [normed_space k F]
include hnfk
Expand All @@ -32,6 +34,7 @@ def bounded_linear_maps : subspace k (E → F) :=
add := assume A B, is_bounded_linear_map.add,
smul := assume c A, is_bounded_linear_map.smul c }

-- introduce the local notation L(E,F) for the set of bounded linear maps.
local notation `L(` E `,` F `)` := @bounded_linear_maps k E F _ _ _

/-- Coerce bounded linear maps to functions. -/
Expand All @@ -41,117 +44,102 @@ instance bounded_linear_maps.to_fun : has_coe_to_fun $ L(E,F) :=
@[extensionality] theorem ext {A B : L(E,F)} (H : ∀ x, A x = B x) : A = B :=
set_coe.ext $ funext H

/-- Bounded linear maps are ... bounded -/
lemma exists_bound (A : L(E,F)) : ∃ c, c > 0 ∧ ∀ x : E, ∥A x∥ ≤ c * ∥x∥ := A.property.bound
def to_linear_map (A : L(E, F)) : linear_map _ E F :=
{to_fun := A.val, ..A.property}

@[simp] lemma bounded_linear_maps.map_zero {A : L(E, F)} : A (0 : E) = 0 :=
(to_linear_map A).map_zero

@[simp] lemma bounded_linear_maps.coe_zero {x : E} : (0 : L(E, F)) x = 0 := rfl

@[simp] lemma bounded_linear_maps.smul_coe {A : L(E, F)} {x : E} {c : k} :
(c • A) x = c • (A x) := rfl

@[simp] lemma bounded_linear_maps.zero_smul {A : L(E, F)} : (0 : k) • A = 0 :=
by ext; simp

/-- Bounded linear maps are conveniently bounded on the unit ball. -/
lemma exists_bound' (A : L(E,F)) : ∃ c, c > 0 ∧ ∀ x : E, ∥x∥ ≤ 1 → ∥A x∥ ≤ c :=
let ⟨c, _, H⟩ := exists_bound A in
exists.intro c ⟨‹c > 0›,
assume x _,
calc ∥A x∥ ≤ c * ∥x∥ : H x
... ≤ c * 1 : (mul_le_mul_left ‹c > 0›).mpr ‹∥x∥ ≤ 1›
... = c : mul_one c⟩
@[simp] lemma bounded_linear_maps.one_smul {A : L(E, F)} : (1 : k) • A = A :=
by ext; simp

/-- Bounded linear maps are ... bounded -/
lemma exists_bound (A : L(E,F)) : ∃ c, 0 < c ∧ ∀ x : E, ∥A x∥ ≤ c * ∥x∥ := A.property.bound

end bounded_linear_maps

/-
Now define the operator norm. We only do this for normed spaces over ℝ, since we need a
scalar multiplication with reals to prove that ∥A x∥ ≤ ∥A∥ * ∥x∥. It would be enough to
have a vector space over a normed field k with a real scalar multiplication and certain
compatibility conditions.
Now define the operator norm.
The main task is to show that the operator norm is definite, homogeneous, and satisfies the
triangle inequality. This is done after a few preliminary lemmas necessary to deal with cSup.
triangle inequality. This is done after a few preliminary lemmas necessary to deal with csupr.
-/
section operator_norm

variables [normed_space ℝ E] [normed_space ℝ F]
variables [normed_field k] [normed_space k E] [normed_space k F]
open lattice set

local notation `L(` E `,` F `)` := @bounded_linear_maps ℝ E F _ _ _

noncomputable def to_linear_map (A : L(E, F)) : linear_map _ E F :=
{to_fun := A.val, ..A.property}
local notation `L(` E `,` F `)` := @bounded_linear_maps k E F _ _ _

/-- The operator norm of a bounded linear map A : E → F is the sup of
the set ∥A x∥ with ∥x∥ ≤ 1. If E = {0} we set ∥A∥ = 0. -/
/-- The operator norm of a bounded linear map A : E → F is the sup of ∥A x∥/∥x∥. -/
noncomputable def operator_norm (A : L(E, F)) : ℝ :=
Sup (image (λ x, ∥A x∥) {x | ∥x∥ ≤ 1})
supr (λ x, ∥A x∥/∥x∥)

noncomputable instance bounded_linear_maps.to_has_norm : has_norm L(E,F) :=
{norm := operator_norm}

lemma norm_of_unit_ball_bdd_above (A : L(E,F)) : bdd_above (image (norm ∘ A) {x | ∥x∥ ≤ 1}) :=
let ⟨c, _, H⟩ := (exists_bound' A : ∃ c, c > 0 ∧ ∀ x : E, ∥x∥ ≤ 1 → ∥A x∥ ≤ c) in
bdd_above.mk c
(assume r ⟨x, (_ : ∥x∥ ≤ 1), (_ : ∥A x∥ = r)⟩,
show r ≤ c, from
calc r = ∥A x∥ : eq.symm ‹∥A x∥ = r›
... ≤ c : H x ‹∥x∥ ≤ 1›)

lemma zero_in_im_ball (A : L(E,F)) : (0:ℝ) ∈ {r : ℝ | ∃ (x : E), ∥x∥ ≤ 1 ∧ ∥A x∥ = r} :=
have A 0 = 0, from (to_linear_map A).map_zero,
exists.intro (0:E) $ and.intro (by rw[norm_zero]; exact zero_le_one) (by rw[‹A 0 = 0›]; simp)
lemma bdd_above_range_norm_image_div_norm (A : L(E,F)) : bdd_above (range (λ x, ∥A x∥/∥x∥)) :=
let ⟨c, _, H⟩ := (exists_bound A : ∃ c, 0 < c ∧ ∀ x : E, ∥A x∥ ≤ c * ∥x∥) in
bdd_above.mk c $ forall_range_iff.2 $ λx, begin
by_cases h : ∥x∥ = 0,
{ simp [h, le_of_lt ‹0 < c›] },
{ refine (div_le_iff _).2 (H x),
simpa [ne.symm h] using le_iff_eq_or_lt.1 (norm_nonneg x) }
end

lemma operator_norm_nonneg (A : L(E,F)) : 0 ≤ ∥A∥ :=
have (0:ℝ) ∈ _, from zero_in_im_ball A,
show 0 ≤ Sup (image (norm ∘ A) {x | ∥x∥ ≤ 1}), from
let ⟨c, _, H⟩ := (exists_bound' A : ∃ c, c > 0 ∧ ∀ x : E, ∥x∥ ≤ 1 → ∥A x∥ ≤ c) in
le_cSup (norm_of_unit_ball_bdd_above A) ‹(0:ℝ) ∈ _›

lemma bounded_by_operator_norm_on_unit_vector (A : L(E, F)) {x : E} (_ : ∥x∥ = 1) : ∥A x∥ ≤ ∥A∥ :=
show ∥A x∥ ≤ Sup (image (norm ∘ A) {x | ∥x∥ ≤ 1}), from
let ⟨c, _, _⟩ := (exists_bound A : ∃ c, c > 0 ∧ ∀ x : E, ∥ A x ∥ ≤ c * ∥ x ∥) in
have ∥A x∥ ∈ (image (norm ∘ A) {x | ∥x∥ ≤ 1}), from mem_image_of_mem _ $ le_of_eq ‹∥x∥ = 1›,
le_cSup (norm_of_unit_ball_bdd_above A) ‹∥A x∥ ∈ _›
begin
have : (0 : ℝ) = (λ x, ∥A x∥/∥x∥) 0, by simp,
rw this,
exact le_csupr (bdd_above_range_norm_image_div_norm A),
end

set_option class.instance_max_depth 34

/-- This is the fundamental property of the operator norm: ∥A x∥ ≤ ∥A∥ * ∥x∥. -/
theorem bounded_by_operator_norm {A : L(E,F)} {x : E} : ∥A x∥ ≤ ∥A∥ * ∥x∥ :=
have A 0 = 0, from (to_linear_map A).map_zero,
classical.by_cases
(assume : x = (0:E),
show ∥A x∥ ≤ ∥A∥ * ∥x∥, by rw[‹x = 0›, ‹A 0 = 0›, norm_zero, norm_zero, mul_zero]; exact le_refl 0)
(assume : x ≠ (0:E),
have ∥x∥ ≠ 0, from ne_of_gt $ (norm_pos_iff x).mpr ‹x ≠ 0›,
have ∥∥x∥⁻¹∥ = ∥x∥⁻¹, from abs_of_nonneg $ inv_nonneg.mpr $ norm_nonneg x,
have ∥∥x∥⁻¹•x∥ = 1, begin rw[norm_smul, ‹∥∥x∥⁻¹∥ = ∥x∥⁻¹›], exact inv_mul_cancel ‹∥x∥ ≠ 0› end,
calc ∥A x∥ = (∥x∥ * ∥x∥⁻¹) * ∥A x∥ : by rw[mul_inv_cancel ‹∥x∥ ≠ 0›]; ring
... = ∥∥x∥⁻¹∥ * ∥A x∥ * ∥x∥ : by rw[‹∥∥x∥⁻¹∥ = ∥x∥⁻¹›]; ring
... = ∥∥x∥⁻¹• A x ∥ * ∥x∥ : by rw[←normed_space.norm_smul ∥x∥⁻¹ (A x)]
... = ∥A (∥x∥⁻¹• x)∥ * ∥x∥ : by {
change ∥∥x∥⁻¹ • A.val x∥ * ∥x∥ = ∥A.val (∥x∥⁻¹ • x)∥ * ∥x∥,
rw A.property.smul}
... ≤ ∥A∥ * ∥x∥ : (mul_le_mul_right ((norm_pos_iff x).mpr ‹x ≠ 0›)).mpr
(bounded_by_operator_norm_on_unit_vector A ‹∥∥x∥⁻¹•x∥ = 1›))

lemma bounded_by_operator_norm_on_unit_ball (A : L(E, F)) {x : E} (_ : ∥x∥ ≤ 1) : ∥A x∥ ≤ ∥A∥ :=
calc ∥A x∥ ≤ ∥A∥ * ∥x∥ : bounded_by_operator_norm
... ≤ ∥A∥ * 1 : mul_le_mul_of_nonneg_left ‹∥x∥ ≤ 1› (operator_norm_nonneg A)
... = ∥A∥ : mul_one ∥A∥

lemma operator_norm_bounded_by {A : L(E,F)} (c : nnreal) :
(∀ x : E, ∥x∥ ≤ 1 → ∥A x∥ ≤ (c:ℝ)) → ∥A∥ ≤ c :=
assume H : ∀ x : E, ∥x∥ ≤ 1 → ∥A x∥ ≤ c,
suffices Sup (image (norm ∘ A) {x | ∥x∥ ≤ 1}) ≤ c, by assumption,
cSup_le (set.ne_empty_of_mem $ zero_in_im_ball A)
(show ∀ (r : ℝ), r ∈ (image (norm ∘ A) {x | ∥x∥ ≤ 1}) → r ≤ c, from
assume r ⟨x, _, _⟩,
calc r = ∥A x∥ : eq.symm ‹_›
... ≤ c : H x ‹_›)

theorem operator_norm_triangle (A : L(E,F)) (B : L(E,F)) : ∥A + B∥ ≤ ∥A∥ + ∥B∥ :=
operator_norm_bounded_by (⟨∥A∥, operator_norm_nonneg A⟩ + ⟨∥B∥, operator_norm_nonneg B⟩)
(assume x _, by exact
begin
classical, by_cases h : x = 0,
{ simp [h] },
{ have : ∥A x∥/∥x∥ ≤ ∥A∥, from le_csupr (bdd_above_range_norm_image_div_norm A),
rwa (div_le_iff _) at this,
have : ∥x∥ ≠ 0, from ne_of_gt ((norm_pos_iff x).mpr h),
have I := le_iff_eq_or_lt.1 (norm_nonneg x),
simpa [ne.symm this] using I }
end

/-- If one controls the norm of every A x, then one controls the norm of A. -/
lemma operator_norm_bounded_by {A : L(E,F)} {c : ℝ} (hc : 0 ≤ c) (H : ∀ x, ∥A x∥ ≤ c * ∥x∥) :
∥A∥ ≤ c :=
begin
haveI : nonempty E := ⟨0⟩,
refine csupr_le (λx, _),
by_cases h : ∥x∥ = 0,
{ simp [h, hc] },
{ refine (div_le_iff _).2 (H x),
simpa [ne.symm h] using le_iff_eq_or_lt.1 (norm_nonneg x) }
end

/-- The operator norm satisfies the triangle inequality. -/
theorem operator_norm_triangle (A B : L(E,F)) : ∥A + B∥ ≤ ∥A∥ + ∥B∥ :=
operator_norm_bounded_by (add_nonneg (operator_norm_nonneg A) (operator_norm_nonneg B))
(assume x,
calc ∥(A + B) x∥ = ∥A x + B x∥ : by refl
... ≤ ∥A x∥ + ∥B x∥ : by exact norm_triangle _ _
... ≤ ∥A∥ + ∥B∥ : by exact add_le_add (bounded_by_operator_norm_on_unit_ball A ‹_›)
(bounded_by_operator_norm_on_unit_ball B ‹_›))
... ≤ ∥A∥ * ∥x∥ + ∥B∥ * ∥x∥ :
add_le_add bounded_by_operator_norm bounded_by_operator_norm
... = (∥A∥ + ∥B∥) * ∥x∥ : by ring)

/-- An operator is zero iff its norm vanishes. -/
theorem operator_norm_zero_iff (A : L(E,F)) : ∥A∥ = 0 ↔ A = 0 :=
have A 0 = 0, from (to_linear_map A).map_zero,
iff.intro
(assume : ∥A∥ = 0,
suffices ∀ x, A x = 0, from ext this,
Expand All @@ -161,62 +149,45 @@ iff.intro
... = 0 : by rw[‹∥A∥ = 0›]; ring,
(norm_le_zero_iff (A x)).mp this)
(assume : A = 0,
let M := {r : ℝ | ∃ (x : E), ∥x∥ ≤ 1 ∧ ∥A x∥ = r} in
-- note that we have M = (image (norm ∘ A) {x | ∥x∥ ≤ 1}), from rfl
suffices Sup M = 0, by assumption,
suffices M = {0}, by rw[this]; exact cSup_singleton 0,
(set.ext_iff M {0}).mpr $ assume r, iff.intro
(assume : ∃ (x : E), ∥x∥ ≤ 1 ∧ ∥A x∥ = r,
let ⟨x, _, _⟩ := this in
have h : ∥(0:F)∥ = r, by rwa[‹A=0›] at *,
by finish)
(assume : r ∈ {0},
have r = 0, from set.eq_of_mem_singleton this,
exists.intro (0:E) $ ⟨by rw[norm_zero]; exact zero_le_one, by rw[this, ‹A 0 = 0›]; simp⟩))

theorem operator_norm_homogeneous (c : ℝ) (A : L(E, F)) : ∥c • A∥ = ∥c∥ * ∥A∥ :=
-- ∥c • A∥ is the supremum of the image of the map x ↦ ∥c • A x∥ on the unit ball in E
-- we show that this is the same as ∥c∥ * ∥A∥ by showing 1) and 2):
-- 1) ∥c∥ * ∥A∥ is an upper bound for the image of x ↦ ∥c • A x∥ on the unit ball
-- 2) any w < ∥c∥ * ∥A∥ is not an upper bound (this is equivalent to showing that every upper bound is ≥ ∥c∥ * ∥A∥)
suffices (∀ a ∈ _, a ≤ ∥c∥ * ∥A∥) ∧ (∀ (ub : ℝ), (∀ a ∈ _, a ≤ ub) → ∥c∥ * ∥A∥ ≤ ub), from
cSup_intro' (show _ ≠ ∅, from set.ne_empty_of_mem $ zero_in_im_ball _) this.1 this.2,
and.intro
(show ∀ a ∈ image (λ x, ∥(c • A) x∥) {x : E | ∥x∥ ≤ 1}, a ≤ ∥c∥ * ∥A∥, from
assume a (hₐ : ∃ (x : E), ∥x∥ ≤ 1 ∧ ∥(c • A) x∥ = a),
let ⟨x, _, _⟩ := hₐ in
calc a = ∥c • A x∥ : eq.symm ‹_›
... = ∥c∥ * ∥A x∥ : by rw[←norm_smul c (A x)]; refl
... ≤ ∥c∥ * ∥A∥ : mul_le_mul_of_nonneg_left
(bounded_by_operator_norm_on_unit_ball A ‹∥x∥ ≤ 1›)
(norm_nonneg c))
(show ∀ (ub : ℝ), (∀ a ∈ image (λ (x : E), ∥(c • A) x∥) {x : E | ∥x∥ ≤ 1}, a ≤ ub) → ∥c∥ * ∥A∥ ≤ ub, from
assume u u_is_ub,
classical.by_cases
(assume : c = 0,
calc ∥c∥ * ∥A∥ = 0 : by rw[‹c=0›, norm_zero, zero_mul]
... ≤ u : u_is_ub (0:ℝ) $ zero_in_im_ball _)
(assume : c ≠ 0,
have ∥c∥ ≠ 0, from ne_of_gt $ (norm_pos_iff c).mpr ‹c ≠ 0›,
have bla : u = ∥c∥ * (∥c∥⁻¹ * u), by rw[←mul_assoc, mul_inv_cancel ‹∥c∥ ≠ 0›, one_mul],
suffices ∥A∥ ≤ ∥c∥⁻¹ * u, from
have u = ∥c∥ * (∥c∥⁻¹ * u), by rw[←mul_assoc, mul_inv_cancel ‹∥c∥ ≠ 0›, one_mul],
by rw[this]; exact mul_le_mul_of_nonneg_left ‹_› (norm_nonneg c),
cSup_le
(set.ne_empty_of_mem $ zero_in_im_ball _)
(assume n (H : ∃ (x : E), ∥x∥ ≤ 1 ∧ ∥A x∥ = n),
let ⟨x, _, _⟩ := H in
calc n = ∥A x∥ : eq.symm ‹∥A x∥ = n›
... = ∥c∥⁻¹ * ∥c • A x∥ : by rw[norm_smul, ←mul_assoc, inv_mul_cancel ‹∥c∥ ≠ 0›, one_mul]
... ≤ ∥c∥⁻¹ * u : mul_le_mul_of_nonneg_left (u_is_ub ∥c • A x∥ ⟨x, ‹∥x∥ ≤ 1›, rfl⟩) $
inv_nonneg.mpr $ norm_nonneg c)))
have ∥A∥ ≤ 0, from operator_norm_bounded_by (le_refl 0) $ by rw this; simp [le_refl],
le_antisymm this (operator_norm_nonneg A))

section instance_70
-- one needs to increase the max_depth for the following proof
set_option class.instance_max_depth 70

/-- Auxiliary lemma to prove that the operator norm is homogeneous. -/
lemma operator_norm_homogeneous_le (c : k) (A : L(E, F)) : ∥c • A∥ ≤ ∥c∥ * ∥A∥ :=
begin
apply operator_norm_bounded_by (mul_nonneg (norm_nonneg _) (operator_norm_nonneg _)) (λx, _),
erw [norm_smul _ _, mul_assoc],
exact mul_le_mul_of_nonneg_left bounded_by_operator_norm (norm_nonneg _)
end

theorem operator_norm_homogeneous (c : k) (A : L(E, F)) : ∥c • A∥ = ∥c∥ * ∥A∥ :=
begin
refine le_antisymm (operator_norm_homogeneous_le c A) _,
by_cases h : ∥c∥ = 0,
{ have : c = 0, from (norm_eq_zero _).1 h,
have I : ∥(0 : L(E, F))∥ = 0, by rw operator_norm_zero_iff,
simp only [h],
simp [this, I] },
{ have h' : c ≠ 0, by simpa [norm_eq_zero] using h,
have : ∥A∥ ≤ ∥c • A∥ / ∥c∥, from calc
∥A∥ = ∥(1 : k) • A∥ : by rw bounded_linear_maps.one_smul
... = ∥c⁻¹ • c • A∥ : by rw [smul_smul, inv_mul_cancel h']
... ≤ ∥c⁻¹∥ * ∥c • A∥ : operator_norm_homogeneous_le _ _
... = ∥c • A∥ / ∥c∥ : by rw [norm_inv, div_eq_inv_mul],
rwa [le_div_iff, mul_comm] at this,
simpa [ne.symm h] using le_iff_eq_or_lt.1 (norm_nonneg c) }
end
end instance_70

/-- Expose L(E,F) equipped with the operator norm as normed space. -/
noncomputable instance bounded_linear_maps.to_normed_space : normed_space ℝ L(E,F) :=
normed_space.of_core ℝ L(E,F) {
noncomputable instance bounded_linear_maps.to_normed_space : normed_space k L(E,F) :=
normed_space.of_core k L(E,F) {
norm_eq_zero_iff := operator_norm_zero_iff,
norm_smul := operator_norm_homogeneous,
triangle := operator_norm_triangle
}
triangle := operator_norm_triangle }

end operator_norm

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