156 lines
5.2 KiB
Text
156 lines
5.2 KiB
Text
/-
|
||
Copyright (c) 2016 Robert Y. Lewis. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Author: Robert Y. Lewis
|
||
|
||
Bounded linear operators
|
||
-/
|
||
import .normed_space .real_limit algebra.module
|
||
open real nat classical
|
||
noncomputable theory
|
||
|
||
namespace analysis
|
||
|
||
section bdd_lin_op
|
||
|
||
structure is_bdd_linear_map [class] {V W : Type} [normed_vector_space V] [normed_vector_space W] (f : V → W) extends is_linear_map ℝ f :=
|
||
(op_norm : ℝ) (op_norm_pos : op_norm > 0) (op_norm_bound : ∀ v : V, ∥f v∥ ≤ op_norm * ∥v∥)
|
||
|
||
/-theorem is_bdd_linear_map_id [instance] (V : Type) [normed_vector_space V] : is_bdd_linear_map (λ a : V, a) :=
|
||
sorry-/
|
||
|
||
theorem is_bdd_linear_map_add [instance] {V W : Type} [normed_vector_space V] [normed_vector_space W]
|
||
(f g : V → W) [Hbf : is_bdd_linear_map f] [Hbg : is_bdd_linear_map g] :
|
||
is_bdd_linear_map (λ x, f x + g x) :=
|
||
begin
|
||
fapply is_bdd_linear_map.mk,
|
||
{intros,
|
||
rewrite [linear_map_additive ℝ f, linear_map_additive ℝ g],
|
||
simp},
|
||
{intros,
|
||
rewrite [linear_map_homogeneous f, linear_map_homogeneous g, smul_left_distrib]},
|
||
{exact is_bdd_linear_map.op_norm _ _ f + is_bdd_linear_map.op_norm _ _ g},
|
||
{apply add_pos,
|
||
repeat apply is_bdd_linear_map.op_norm_pos},
|
||
{intro,
|
||
apply le.trans,
|
||
apply norm_triangle,
|
||
rewrite right_distrib,
|
||
apply add_le_add,
|
||
repeat apply is_bdd_linear_map.op_norm_bound}
|
||
end
|
||
|
||
variables {V W : Type}
|
||
variables [HV : normed_vector_space V] [HW : normed_vector_space W]
|
||
--variable f : V → W --linear_operator V W
|
||
include HV HW
|
||
variable f : V → W
|
||
variable [Hf : is_bdd_linear_map f]
|
||
include Hf
|
||
|
||
definition op_norm := is_bdd_linear_map.op_norm _ _ f
|
||
|
||
theorem op_norm_pos : op_norm f > 0 := is_bdd_linear_map.op_norm_pos _ _ f
|
||
|
||
theorem op_norm_bound (v : V) : ∥f v∥ ≤ (op_norm f) * ∥v∥ := is_bdd_linear_map.op_norm_bound _ _ f v
|
||
|
||
theorem bounded_linear_operator_continuous : continuous f :=
|
||
begin
|
||
intro x,
|
||
apply normed_vector_space.continuous_at_intro,
|
||
intro ε Hε,
|
||
existsi ε / (op_norm f),
|
||
split,
|
||
apply div_pos_of_pos_of_pos Hε !op_norm_pos,
|
||
intro x' Hx',
|
||
rewrite [-linear_map_sub f],
|
||
apply lt_of_le_of_lt,
|
||
apply op_norm_bound f,
|
||
rewrite [-@mul_div_cancel' _ _ ε (op_norm f) (ne_of_gt !op_norm_pos)],
|
||
apply mul_lt_mul_of_pos_left,
|
||
exact Hx',
|
||
apply op_norm_pos
|
||
end
|
||
|
||
end bdd_lin_op
|
||
|
||
section frechet_deriv
|
||
variables {V W : Type}
|
||
variables [HV : normed_vector_space V] [HW : normed_vector_space W]
|
||
include HV HW
|
||
|
||
open topology
|
||
|
||
definition is_frechet_deriv_at (f A : V → W) [is_bdd_linear_map A] (x : V) :=
|
||
(λ h : V, ∥f (x + h) - f x - A h ∥ / ∥ h ∥) ⟶ 0 at 0
|
||
|
||
structure frechet_diffable_at [class] (f : V → W) (x : V) :=
|
||
(A : V → W) [HA : is_bdd_linear_map A] (is_fr_der : is_frechet_deriv_at f A x)
|
||
|
||
variables f g : V → W
|
||
variable x : V
|
||
|
||
definition frechet_deriv_at [Hf : frechet_diffable_at f x] : V → W :=
|
||
frechet_diffable_at.A _ _ f x
|
||
|
||
definition frechet_deriv_at_is_bdd_linear_map [instance] (f : V → W) (x : V) [Hf : frechet_diffable_at f x] :
|
||
is_bdd_linear_map (frechet_deriv_at f x) :=
|
||
frechet_diffable_at.HA _ _ f x
|
||
|
||
theorem frechet_deriv_spec [Hf : frechet_diffable_at f x] :
|
||
(λ h : V, ∥f (x + h) - f x - (frechet_deriv_at f x h) ∥ / ∥ h ∥) ⟶ 0 at 0 :=
|
||
frechet_diffable_at.is_fr_der _ _ f x
|
||
|
||
theorem frechet_diffable_at_add (A B : V → W) [is_bdd_linear_map A] [is_bdd_linear_map B]
|
||
(Hf : is_frechet_deriv_at f A x) (Hg : is_frechet_deriv_at g B x) :
|
||
is_frechet_deriv_at (λ y, f y + g y) (λ y, A y + B y) x :=
|
||
begin
|
||
rewrite ↑is_frechet_deriv_at,
|
||
have Hle : ∀ h, ∥f (x + h) + g (x + h) - (f x + g x) - (A h + B h)∥ / ∥h∥ ≤
|
||
∥f (x + h) - f x - A h∥ / ∥h∥ + ∥g (x + h) - g x - B h∥ / ∥h∥, begin
|
||
intro h,
|
||
cases em (∥h∥ > 0) with Hh Hh,
|
||
krewrite div_add_div_same,
|
||
apply div_le_div_of_le_of_pos,
|
||
have Hfeq : f (x + h) + g (x + h) - (f x + g x) - (A h + B h) =
|
||
(f (x + h) - f x - A h) + (g (x + h) - g x - B h), by simp,
|
||
rewrite Hfeq,
|
||
apply norm_triangle,
|
||
exact Hh,
|
||
have Hhe : ∥h∥ = 0, from eq_of_le_of_ge (le_of_not_gt Hh) !norm_nonneg,
|
||
krewrite [Hhe, *div_zero, zero_add],
|
||
eapply le.refl
|
||
end,
|
||
have Hlimge : (λ h, ∥f (x + h) - f x - A h∥ / ∥h∥ + ∥g (x + h) - g x - B h∥ / ∥h∥) ⟶ 0 at 0, begin
|
||
rewrite [-zero_add 0],
|
||
apply add_converges_to_at,
|
||
apply Hf,
|
||
apply Hg
|
||
end,
|
||
have Hlimle : (λ (h : V), (0 : ℝ)) ⟶ 0 at 0, from converges_to_at_constant 0 0,
|
||
apply converges_to_at_squeeze Hlimle Hlimge,
|
||
intro y,
|
||
apply div_nonneg_of_nonneg_of_nonneg,
|
||
repeat apply norm_nonneg,
|
||
apply Hle
|
||
end
|
||
|
||
/-theorem continuous_at_of_diffable_at [Hf : frechet_diffable_at f x] : continuous_at f x :=
|
||
begin
|
||
apply normed_vector_space.continuous_at_intro,
|
||
intros ε Hε,
|
||
note Hfds := frechet_deriv_spec f x Hε,
|
||
cases Hfds with δ Hδ,
|
||
cases Hδ with Hδ Hδ',
|
||
existsi δ,
|
||
split,
|
||
assumption,
|
||
intro x' Hx',
|
||
have Hx'x : x' - x ≠ 0 ∧ dist (x' - x) 0 < δ, from sorry,
|
||
note Hδ'' := Hδ' Hx'x,
|
||
|
||
end-/
|
||
|
||
end frechet_deriv
|
||
|
||
end analysis
|