feat(theories/analysis): define frechet derivative + basic theorems

library/theories/analysis/bounded_linear_operator.lean

@@ -0,0 +1,156 @@
+/-
+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

library/theories/analysis/metric_space.lean

@@ -399,6 +399,17 @@ definition converges_to_at (f : M → N) (y : N) (x : M) :=
 
 notation f `⟶` y `at` x := converges_to_at f y x
 
+theorem converges_to_at_constant (y : N) (x : M) : (λ m, y) ⟶ y at x :=
+  begin
+    intros ε Hε,
+    existsi 1,
+    split,
+    exact zero_lt_one,
+    intros x' Hx',
+    rewrite dist_self,
+    apply Hε
+  end
+
 definition converges_at [class] (f : M → N) (x : M) :=
 ∃ y, converges_to_at f y x
 

library/theories/analysis/normed_space.lean

@@ -241,3 +241,36 @@ proposition norm_converges_to_seq_zero_iff (X : ℕ → V) :
 iff.intro converges_to_seq_zero_of_norm_converges_to_seq_zero norm_converges_to_seq_zero
 
 end analysis
+
+namespace analysis
+variables {U V : Type}
+variable [HU : normed_vector_space U]
+variable [HV : normed_vector_space V]
+variables f g : U → V
+
+include HU HV
+
+theorem add_converges_to_at {lf lg : V} {x : U} (Hf : f ⟶ lf at x) (Hg : g ⟶ lg at x) :
+        (λ y, f y + g y) ⟶ lf + lg at x :=
+  begin
+    apply converges_to_at_of_all_conv_seqs,
+    intro X HX,
+    apply add_converges_to_seq,
+    apply all_conv_seqs_of_converges_to_at Hf,
+    apply HX,
+    apply all_conv_seqs_of_converges_to_at Hg,
+    apply HX
+  end
+
+open topology
+
+theorem normed_vector_space.continuous_at_intro {x : U}
+        (H : ∀ ε : ℝ, ε > 0 → (∃ δ : ℝ, δ > 0 ∧ ∀ x' : U, ∥x' - x∥ < δ → ∥f x' - f x∥ < ε)) :
+        continuous_at f x :=
+  continuous_at_intro H
+
+theorem normed_vector_space.continuous_at_elim {x : U} (H : continuous_at f x) :
+         ∀ ε : ℝ, ε > 0 → (∃ δ : ℝ, δ > 0 ∧ ∀ x' : U, ∥x' - x∥ < δ → ∥f x' - f x∥ < ε) :=
+  continuous_at_elim H
+
+end analysis