working...

Nevanlinna/bilinear.lean

@@ -1,172 +1,89 @@
---import Mathlib.Algebra.BigOperators.Basic
+/-
-import Mathlib.Analysis.InnerProductSpace.Basic
+Copyright (c) 2024 Stefan Kebekus. All rights reserved.
-import Mathlib.Analysis.InnerProductSpace.Dual
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Stefan Kebekus
+-/
+
 import Mathlib.Analysis.InnerProductSpace.PiL2
 
+/-!
 
-open BigOperators
+# Canoncial Elements in Tensor Powers of Real Inner Product Spaces
-open Finset
 
-variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
+Given an `InnerProductSpace ℝ E`, this file defines two canonical tensors, which
-variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+are relevant when for a coordinate-free definition of differential operators
-open BigOperators
+such as the Laplacian.
-open Finset
+
+* `InnerProductSpace.canonicalContravariantTensor E : E ⊗[ℝ] E →ₗ[ℝ] ℝ`. This is
+  the element corresponding to the inner product.
+
+* If `E` is finite-dimensional, then `E ⊗[ℝ] E` is canonically isomorphic to its
+  dual. Accordingly, there exists an element
+  `InnerProductSpace.canonicalCovariantTensor E : E ⊗[ℝ] E` that corresponds to
+  `InnerProductSpace.canonicalContravariantTensor E` under this identification.
+
+The theorem `InnerProductSpace.canonicalCovariantTensorRepresentation` shows
+that `InnerProductSpace.canonicalCovariantTensor E` can be computed from any
+orthonormal basis `v` as `∑ i, (v i) ⊗ₜ[ℝ] (v i)`.
+
+-/
 
 
+open TensorProduct
 
-lemma vectorPresentation
+
+lemma OrthonormalBasis.sum_repr'
+  {𝕜 : Type*} [RCLike 𝕜]
+  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
   [Fintype ι]
-  (b : Basis ι ℝ E)
+  (b : OrthonormalBasis ι 𝕜 E)
-  (hb : Orthonormal ℝ b)
   (v : E) :
-  v = ∑ i, ⟪b i, v⟫_ℝ • (b i) := by
+  v = ∑ i, ⟪b i, v⟫_𝕜 • (b i) := by
   nth_rw 1 [← (b.sum_repr v)]
-  apply Fintype.sum_congr
+  simp_rw [b.repr_apply_apply v]
-  intro i
-  rw [← Orthonormal.inner_right_finsupp hb (b.repr v) i]
-  simp
 
 
-theorem BilinearCalc
+noncomputable def InnerProductSpace.canonicalContravariantTensor
+  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  : E ⊗[ℝ] E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
+
+
+noncomputable def InnerProductSpace.canonicalCovariantTensor
+  (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] 
+  : E ⊗[ℝ] E := ∑ i, ((stdOrthonormalBasis ℝ E) i) ⊗ₜ[ℝ] ((stdOrthonormalBasis ℝ E) i)
+
+
+theorem InnerProductSpace.canonicalCovariantTensorRepresentation
+  (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
   [Fintype ι]
-  (v : Basis ι ℝ E)
+  (v : OrthonormalBasis ι ℝ E)
-  (c : ι → ℝ)
+  : InnerProductSpace.canonicalCovariantTensor E = ∑ i, (v i) ⊗ₜ[ℝ] (v i) := by
-  (L : E →ₗ[ℝ] E →ₗ[ℝ] F) 
-  : L (∑ j : ι, c j • v j) (∑ j : ι, c j • v j) 
-    = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L (v (x 0)) (v (x 1)) := by
-  
-  rw [map_sum]
-  rw [map_sum]
-
-  conv =>
-    left
-    arg 2
-    intro r
-    rw [← sum_apply]
-
-    rw [map_smul]
-    arg 2
-    arg 1
-    arg 2
-    intro x
-    rw [map_smul]
-  simp
-
-
-lemma c2
-  [Fintype ι]
-  (b : Basis ι ℝ E)
-  (hb : Orthonormal ℝ b)
-  (x y : E) :
-  ⟪x, y⟫_ℝ = ∑ i : ι, ⟪x, b i⟫_ℝ * ⟪y, b i⟫_ℝ := by
-  rw [vectorPresentation b hb x]
-  rw [vectorPresentation b hb y]
-  rw [Orthonormal.inner_sum hb]
-  simp
-  conv =>
-    right
-    arg 2
-    intro i'
-    rw [Orthonormal.inner_left_fintype hb]
-    rw [Orthonormal.inner_left_fintype hb]
-
-
-lemma fin_sum
-  [Fintype ι]
-  (f : ι → ι → F) :
-  ∑ r : Fin 2 → ι, f (r 0) (r 1) = ∑ r₀ : ι, (∑ r₁ : ι, f r₀ r₁) := by
-
-  rw [← Fintype.sum_prod_type']
-  apply Fintype.sum_equiv (finTwoArrowEquiv ι)
-  intro x
-  dsimp 
-
-theorem TensorIndep
-  [Fintype ι] [DecidableEq ι]
-  (v₁ : Basis ι ℝ E)
-  (hv₁ : Orthonormal ℝ v₁)
-  (v₂ : Basis ι ℝ E)
-  (hv₂ : Orthonormal ℝ v₂) :
-  ∑ i, (v₁ i) ⊗ₜ[ℝ] (v₁ i) = ∑ i, (v₂ i) ⊗ₜ[ℝ] (v₂ i) := by
 
+  let w := stdOrthonormalBasis ℝ E
   conv =>
     right
     arg 2
     intro i
-    rw [vectorPresentation v₁ hv₁ (v₂ i)]
+    rw [w.sum_repr' (v i)]
-    rw [TensorProduct.sum_tmul]
+  simp_rw [TensorProduct.sum_tmul, TensorProduct.tmul_sum, TensorProduct.smul_tmul_smul]
-    arg 2
-    intro j
-    rw [TensorProduct.tmul_sum]
-    arg 2
-    intro a
-    rw [TensorProduct.tmul_smul]
-    arg 2
-    rw [TensorProduct.smul_tmul]
-
-  rw [Finset.sum_comm]
-  conv =>
-    right
-    arg 2
-    intro i
-    rw [Finset.sum_comm]
-
-
-
-  sorry
-
-theorem LaplaceIndep
-  [Fintype ι] [DecidableEq ι]
-  (v₁ : Basis ι ℝ E)
-  (hv₁ : Orthonormal ℝ v₁)
-  (v₂ : Basis ι ℝ E)
-  (hv₂ : Orthonormal ℝ v₂)
-  (L : E →ₗ[ℝ] E →ₗ[ℝ] F) :
-  ∑ i, L (v₁ i) (v₁ i) = ∑ i, L (v₂ i) (v₂ i) := by
-
-  have vector_vs_function
-    {y : Fin 2 → ι} 
-    {v : ι → E}
-    : (fun i => v (y i)) = ![v (y 0), v (y 1)] := by 
-    funext i
-    by_cases h : i = 0
-    · rw [h]
-      simp
-    · rw [Fin.eq_one_of_neq_zero i h]
-      simp
   
   conv =>
     right
-    arg 2
-    intro i
-    rw [vectorPresentation v₁ hv₁ (v₂ i)]
-    rw [BilinearCalc]
     rw [Finset.sum_comm]
-  conv =>
-    right
     arg 2
     intro y
-    rw [← Finset.sum_smul]
+    rw [Finset.sum_comm]
-    rw [← c2 v₂ hv₂ (v₁ (y 0)) (v₁ (y 1))]
-    rw [vector_vs_function]
-  simp
-
-  rw [fin_sum (fun i₀ ↦ (fun i₁ ↦ ⟪v₁ i₀, v₁ i₁⟫_ℝ • L ![v₁ i₀, v₁ i₁]))]
-    
-  have xx {r₀ : ι} : ∀ r₁ : ι, r₁ ≠ r₀ → ⟪v₁ r₀, v₁ r₁⟫_ℝ • L ![v₁ r₀, v₁ r₁] = 0 := by
-    intro r₁ hr₁
-    rw [orthonormal_iff_ite.1 hv₁]
-    simp
-    tauto
-
-  conv =>
-    right
     arg 2
-    intro r₀
+    intro x
-    rw [Fintype.sum_eq_single r₀ xx]
+    rw [← Finset.sum_smul]
-    rw [orthonormal_iff_ite.1 hv₁]
+    arg 1
-  apply sum_congr
+    arg 2
+    intro i
+    rw [← real_inner_comm (w x)]
+  simp_rw [OrthonormalBasis.sum_inner_mul_inner v]
+
+  have {i} : ∑ j, ⟪w i, w j⟫_ℝ • w i ⊗ₜ[ℝ] w j = w i ⊗ₜ[ℝ] w i := by
+    rw [Fintype.sum_eq_single i, orthonormal_iff_ite.1 w.orthonormal]; simp
+    intro _ _; rw [orthonormal_iff_ite.1 w.orthonormal]; simp; tauto
+  simp_rw [this]
   rfl
-  intro x _
-  rw [vector_vs_function]
-  simp

Nevanlinna/laplace2.lean

@@ -10,7 +10,6 @@ import Mathlib.LinearAlgebra.Contraction
 open BigOperators
 open Finset
 
-
 lemma OrthonormalBasis.sum_repr'
   {𝕜 : Type*} [RCLike 𝕜]
   {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
@@ -21,15 +20,53 @@ lemma OrthonormalBasis.sum_repr'
   nth_rw 1 [← (b.sum_repr v)]
   simp_rw [b.repr_apply_apply v]
 
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
+
 noncomputable def realInnerAsElementOfDualTensorprod
   {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
   : TensorProduct ℝ E E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
 
+
+instance 
+  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]  [FiniteDimensional ℝ E]
+  : NormedAddCommGroup (TensorProduct ℝ E E) where 
+    norm := by 
+      
+      sorry
+    dist_self := by 
+      
+      sorry
+
+  sorry
+
+/-
+instance 
+  {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]  [FiniteDimensional ℝ E]
+  : InnerProductSpace ℝ (TensorProduct ℝ E E) where
+    smul := _
+    one_smul := _
+    mul_smul := _
+    smul_zero := _
+    smul_add := _
+    add_smul := _
+    zero_smul := _
+    norm_smul_le := _
+    inner := _
+    norm_sq_eq_inner := _
+    conj_symm := _
+    add_left := _
+    smul_left := _
+  -/
+
+
+
 noncomputable def dual'
   {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]  [FiniteDimensional ℝ E]
   : (TensorProduct ℝ E E →ₗ[ℝ] ℝ) ≃ₗ[ℝ] TensorProduct ℝ E E := by
 
   let d := InnerProductSpace.toDual ℝ E
+
+
   let e := d.toLinearEquiv
 
   let a := TensorProduct.congr e e