116 lines
4.2 KiB
Text
116 lines
4.2 KiB
Text
/-
|
||
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Author: Jeremy Avigad
|
||
|
||
Normed spaces.
|
||
-/
|
||
import algebra.module .real_limit
|
||
open real
|
||
|
||
noncomputable theory
|
||
|
||
structure has_norm [class] (M : Type) : Type :=
|
||
(norm : M → ℝ)
|
||
|
||
definition norm {M : Type} [has_normM : has_norm M] (v : M) : ℝ := has_norm.norm v
|
||
|
||
notation `∥`v`∥` := norm v
|
||
|
||
-- where is the right place to put this?
|
||
structure real_vector_space [class] (V : Type) extends vector_space ℝ V
|
||
|
||
structure normed_vector_space [class] (V : Type) extends real_vector_space V, has_norm V :=
|
||
(norm_zero : norm zero = 0)
|
||
(eq_zero_of_norm_eq_zero : ∀ u : V, norm u = 0 → u = zero)
|
||
(norm_triangle : ∀ u v, norm (add u v) ≤ norm u + norm v)
|
||
(norm_smul : ∀ (a : ℝ) (v : V), norm (smul a v) = abs a * norm v)
|
||
|
||
-- what namespace should we put this in?
|
||
|
||
section normed_vector_space
|
||
variable {V : Type}
|
||
variable [normed_vector_space V]
|
||
|
||
proposition norm_zero : ∥ (0 : V) ∥ = 0 := !normed_vector_space.norm_zero
|
||
|
||
proposition eq_zero_of_norm_eq_zero {u : V} (H : ∥ u ∥ = 0) : u = 0 :=
|
||
!normed_vector_space.eq_zero_of_norm_eq_zero H
|
||
|
||
proposition norm_triangle (u v : V) : ∥ u + v ∥ ≤ ∥ u ∥ + ∥ v ∥ :=
|
||
!normed_vector_space.norm_triangle
|
||
|
||
proposition norm_smul (a : ℝ) (v : V) : ∥ a • v ∥ = abs a * ∥ v ∥ :=
|
||
!normed_vector_space.norm_smul
|
||
|
||
proposition norm_neg (v : V) : ∥ -v ∥ = ∥ v ∥ :=
|
||
have abs (1 : ℝ) = 1, from abs_of_nonneg zero_le_one,
|
||
by+ rewrite [-@neg_one_smul ℝ V, norm_smul, abs_neg, this, one_mul]
|
||
|
||
section
|
||
private definition nvs_dist (u v : V) := ∥ u - v ∥
|
||
|
||
private lemma nvs_dist_self (u : V) : nvs_dist u u = 0 :=
|
||
by rewrite [↑nvs_dist, sub_self, norm_zero]
|
||
|
||
private lemma eq_of_nvs_dist_eq_zero (u v : V) (H : nvs_dist u v = 0) : u = v :=
|
||
have u - v = 0, from eq_zero_of_norm_eq_zero H,
|
||
eq_of_sub_eq_zero this
|
||
|
||
private lemma nvs_dist_triangle (u v w : V) : nvs_dist u w ≤ nvs_dist u v + nvs_dist v w :=
|
||
calc
|
||
nvs_dist u w = ∥ (u - v) + (v - w) ∥ : by rewrite [↑nvs_dist, *sub_eq_add_neg, add.assoc,
|
||
neg_add_cancel_left]
|
||
... ≤ ∥ u - v ∥ + ∥ v - w ∥ : norm_triangle
|
||
|
||
private lemma nvs_dist_comm (u v : V) : nvs_dist u v = nvs_dist v u :=
|
||
by rewrite [↑nvs_dist, -norm_neg, neg_sub]
|
||
end
|
||
|
||
definition normed_vector_space_to_metric_space [reducible] [trans_instance] : metric_space V :=
|
||
⦃ metric_space,
|
||
dist := nvs_dist,
|
||
dist_self := nvs_dist_self,
|
||
eq_of_dist_eq_zero := eq_of_nvs_dist_eq_zero,
|
||
dist_comm := nvs_dist_comm,
|
||
dist_triangle := nvs_dist_triangle
|
||
⦄
|
||
end normed_vector_space
|
||
|
||
structure banach_space [class] (V : Type) extends nvsV : normed_vector_space V :=
|
||
(complete : ∀ X, @metric_space.cauchy V (@normed_vector_space_to_metric_space V nvsV) X →
|
||
@metric_space.converges_seq V (@normed_vector_space_to_metric_space V nvsV) X)
|
||
|
||
definition banach_space_to_metric_space [reducible] [trans_instance] (V : Type) [bsV : banach_space V] :
|
||
complete_metric_space V :=
|
||
⦃ complete_metric_space, normed_vector_space_to_metric_space,
|
||
complete := banach_space.complete
|
||
⦄
|
||
|
||
section
|
||
open metric_space
|
||
|
||
example (V : Type) (vsV : banach_space V) (X : ℕ → V) (H : cauchy X) : converges_seq X :=
|
||
complete V H
|
||
end
|
||
|
||
/- the real numbers themselves can be viewed as a banach space -/
|
||
|
||
definition real_is_real_vector_space [trans_instance] [reducible] : real_vector_space ℝ :=
|
||
⦃ real_vector_space, real.discrete_linear_ordered_field,
|
||
smul := mul,
|
||
smul_left_distrib := left_distrib,
|
||
smul_right_distrib := right_distrib,
|
||
smul_mul := mul.assoc,
|
||
one_smul := one_mul
|
||
⦄
|
||
|
||
definition real_is_banach_space [trans_instance] [reducible] : banach_space ℝ :=
|
||
⦃ banach_space, real_is_real_vector_space,
|
||
norm := abs,
|
||
norm_zero := abs_zero,
|
||
eq_zero_of_norm_eq_zero := λ a H, eq_zero_of_abs_eq_zero H,
|
||
norm_triangle := abs_add_le_abs_add_abs,
|
||
norm_smul := abs_mul,
|
||
complete := λ X H, complete ℝ H
|
||
⦄
|