258 lines
6.5 KiB
Plaintext
258 lines
6.5 KiB
Plaintext
import Mathlib.Analysis.InnerProductSpace.Basic
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import Mathlib.Analysis.InnerProductSpace.Dual
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import Mathlib.Analysis.InnerProductSpace.PiL2
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--import Mathlib.Algebra.BigOperators.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Bounds
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import Mathlib.Analysis.Calculus.FDeriv.Symmetric
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import Mathlib.LinearAlgebra.Basis
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import Mathlib.LinearAlgebra.Contraction
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open BigOperators
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open Finset
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lemma OrthonormalBasis.sum_repr'
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{𝕜 : Type*} [RCLike 𝕜]
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
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[Fintype ι]
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(b : OrthonormalBasis ι 𝕜 E)
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(v : E) :
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v = ∑ i, ⟪b i, v⟫_𝕜 • (b i) := by
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nth_rw 1 [← (b.sum_repr v)]
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simp_rw [b.repr_apply_apply v]
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noncomputable def realInnerAsElementOfDualTensorprod
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
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: TensorProduct ℝ E E →ₗ[ℝ] ℝ := TensorProduct.lift bilinFormOfRealInner
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noncomputable def dual'
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{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]
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: (TensorProduct ℝ E E →ₗ[ℝ] ℝ) ≃ₗ[ℝ] TensorProduct ℝ E E := by
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let d := InnerProductSpace.toDual ℝ E
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let e := d.toLinearEquiv
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let a := TensorProduct.congr e e
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let b := homTensorHomEquiv ℝ E E ℝ ℝ
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sorry
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variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
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theorem InvariantTensor
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[Fintype ι]
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(v : Basis ι ℝ E)
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(c : ι → ℝ)
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(L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
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L (fun _ => ∑ j : ι, c j • v j) = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L ((fun i => v (x i))) := by
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rw [L.map_sum]
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simp_rw [L.map_smul_univ]
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sorry
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theorem BilinearCalc
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[Fintype ι]
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(v : Basis ι ℝ E)
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(c : ι → ℝ)
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(L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
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L (fun _ => ∑ j : ι, c j • v j) = ∑ x : Fin 2 → ι, (c (x 0) * c (x 1)) • L ((fun i => v (x i))) := by
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rw [L.map_sum]
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simp_rw [L.map_smul_univ]
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sorry
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/-
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lemma c2
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[Fintype ι]
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(b : Basis ι ℝ E)
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(hb : Orthonormal ℝ b)
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(x y : E) :
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⟪x, y⟫_ℝ = ∑ i : ι, ⟪x, b i⟫_ℝ * ⟪y, b i⟫_ℝ := by
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rw [vectorPresentation b hb x]
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rw [vectorPresentation b hb y]
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rw [Orthonormal.inner_sum hb]
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simp
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conv =>
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right
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arg 2
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intro i'
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rw [Orthonormal.inner_left_fintype hb]
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rw [Orthonormal.inner_left_fintype hb]
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-/
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lemma fin_sum
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[Fintype ι]
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(f : ι → ι → F) :
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∑ r : Fin 2 → ι, f (r 0) (r 1) = ∑ r₀ : ι, (∑ r₁ : ι, f r₀ r₁) := by
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rw [← Fintype.sum_prod_type']
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apply Fintype.sum_equiv (finTwoArrowEquiv ι)
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intro x
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dsimp
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theorem LaplaceIndep
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[Fintype ι] [DecidableEq ι]
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(v₁ : OrthonormalBasis ι ℝ E)
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(v₂ : OrthonormalBasis ι ℝ E)
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(L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F) :
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∑ i, L (fun _ ↦ v₁ i) = ∑ i, L (fun _ => v₂ i) := by
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have vector_vs_function
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{y : Fin 2 → ι}
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{v : ι → E}
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: (fun i => v (y i)) = ![v (y 0), v (y 1)] := by
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funext i
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by_cases h : i = 0
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· rw [h]
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simp
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· rw [Fin.eq_one_of_neq_zero i h]
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simp
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conv =>
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right
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arg 2
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intro i
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rw [v₁.sum_repr' (v₂ i)]
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rw [BilinearCalc]
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rw [Finset.sum_comm]
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conv =>
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right
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arg 2
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intro y
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rw [← Finset.sum_smul]
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rw [← c2 v₂ hv₂ (v₁ (y 0)) (v₁ (y 1))]
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rw [vector_vs_function]
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simp
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rw [fin_sum (fun i₀ ↦ (fun i₁ ↦ ⟪v₁ i₀, v₁ i₁⟫_ℝ • L ![v₁ i₀, v₁ i₁]))]
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have xx {r₀ : ι} : ∀ r₁ : ι, r₁ ≠ r₀ → ⟪v₁ r₀, v₁ r₁⟫_ℝ • L ![v₁ r₀, v₁ r₁] = 0 := by
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intro r₁ hr₁
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rw [orthonormal_iff_ite.1 hv₁]
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simp
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tauto
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conv =>
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right
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arg 2
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intro r₀
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rw [Fintype.sum_eq_single r₀ xx]
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rw [orthonormal_iff_ite.1 hv₁]
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apply sum_congr
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rfl
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intro x _
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rw [vector_vs_function]
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simp
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noncomputable def Laplace_wrt_basis
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[Fintype ι]
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(v : Basis ι ℝ E)
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(_ : Orthonormal ℝ v)
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(f : E → F) :
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E → F :=
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fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![v i, v i]
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theorem LaplaceIndep'
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[Fintype ι] [DecidableEq ι]
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(v₁ : OrthonormalBasis ι ℝ E)
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(v₂ : OrthonormalBasis ι ℝ E)
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(f : E → F)
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: (Laplace_wrt_basis v₁ hv₁ f) = (Laplace_wrt_basis v₂ hv₂ f) := by
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funext z
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unfold Laplace_wrt_basis
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let XX := LaplaceIndep v₁ hv₁ v₂ hv₂ (iteratedFDeriv ℝ 2 f z)
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have vector_vs_function
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{v : E}
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: ![v, v] = (fun _ => v) := by
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funext i
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by_cases h : i = 0
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· rw [h]
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simp
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· rw [Fin.eq_one_of_neq_zero i h]
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simp
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conv =>
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left
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arg 2
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intro i
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rw [vector_vs_function]
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conv =>
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right
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arg 2
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intro i
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rw [vector_vs_function]
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assumption
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theorem LaplaceIndep''
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[Fintype ι₁] [DecidableEq ι₁]
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(v₁ : Basis ι₁ ℝ E)
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(hv₁ : Orthonormal ℝ v₁)
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[Fintype ι₂] [DecidableEq ι₂]
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(v₂ : Basis ι₂ ℝ E)
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(hv₂ : Orthonormal ℝ v₂)
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(f : E → F)
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: (Laplace_wrt_basis v₁ hv₁ f) = (Laplace_wrt_basis v₂ hv₂ f) := by
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have b : ι₁ ≃ ι₂ := by
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apply Fintype.equivOfCardEq
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rw [← FiniteDimensional.finrank_eq_card_basis v₁]
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rw [← FiniteDimensional.finrank_eq_card_basis v₂]
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let v'₁ := Basis.reindex v₁ b
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have hv'₁ : Orthonormal ℝ v'₁ := by
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let A := Basis.reindex_apply v₁ b
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have : ⇑v'₁ = v₁ ∘ b.symm := by
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funext i
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exact A i
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rw [this]
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let B := Orthonormal.comp hv₁ b.symm
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apply B
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exact Equiv.injective b.symm
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rw [← LaplaceIndep' v'₁ hv'₁ v₂ hv₂ f]
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unfold Laplace_wrt_basis
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simp
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funext z
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rw [← Equiv.sum_comp b.symm]
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apply Fintype.sum_congr
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intro i₂
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congr
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rw [Basis.reindex_apply v₁ b i₂]
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noncomputable def Laplace
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(f : E → F)
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: E → F := by
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exact Laplace_wrt_basis (stdOrthonormalBasis ℝ E).toBasis (stdOrthonormalBasis ℝ E).orthonormal f
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theorem LaplaceIndep'''
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[Fintype ι] [DecidableEq ι]
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(v : Basis ι ℝ E)
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(hv : Orthonormal ℝ v)
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(f : E → F)
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: (Laplace f) = (Laplace_wrt_basis v hv f) := by
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unfold Laplace
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apply LaplaceIndep'' (stdOrthonormalBasis ℝ E).toBasis (stdOrthonormalBasis ℝ E).orthonormal v hv f
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theorem Complex.Laplace'
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(f : ℂ → F)
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: (Laplace f) = fun z ↦ (iteratedFDeriv ℝ 2 f z) ![1, 1] + (iteratedFDeriv ℝ 2 f z) ![Complex.I, Complex.I] := by
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rw [LaplaceIndep''' Complex.orthonormalBasisOneI.toBasis Complex.orthonormalBasisOneI.orthonormal f]
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unfold Laplace_wrt_basis
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simp
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