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Update laplace2.lean

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Stefan Kebekus 2024-06-24 12:14:18 +02:00
parent ecdc182f2b
commit 42cf2e41b9
1 changed files with 29 additions and 20 deletions
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Nevanlinna/laplace2.lean
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@ -3,6 +3,7 @@ import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.BigOperators.Basic import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Analysis.Calculus.ContDiff.Bounds import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.FDeriv.Symmetric import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.LinearAlgebra.Basis
open BigOperators open BigOperators
open Finset open Finset
@ -10,24 +11,19 @@ open Finset
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
#check EuclideanSpace.norm_eq
#check EuclideanSpace.dist_eq
lemma vectorPresentation
noncomputable def Laplace₁ (n : ℕ) (f : EuclideanSpace ℝ (Fin n) → F) : EuclideanSpace ℝ (Fin n) → F := by
let e : Fin n → EuclideanSpace ℝ (Fin n) := fun i ↦ EuclideanSpace.single i (1 : ℝ)
exact fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![e i, e i]
noncomputable def Laplace₂
[Fintype ι] [Fintype ι]
(v : Basis ι ℝ E) (b : Basis ι ℝ E)
(hv : Orthonormal ℝ v) (hb : Orthonormal ℝ b)
(f : E → F) : (v : E) :
E → F := v = ∑ i, ⟪b i, v⟫_ℝ • (b i) := by
fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![v i, v i] nth_rw 1 [← (b.sum_repr v)]
apply Fintype.sum_congr
intro i
rw [← Orthonormal.inner_right_finsupp hb (b.repr v) i]
simp
#check ContinuousMultilinearMap.map_sum_finset
theorem LaplaceIndep theorem LaplaceIndep
[Fintype ι] [Fintype ι]
@ -54,12 +50,12 @@ theorem LaplaceIndep
--rw [ContinuousMultilinearMap.map_sum_finset] --rw [ContinuousMultilinearMap.map_sum_finset]
have v : E := by sorry have v : E := by sorry
let t := ![∑ j, ⟪v₁ j, v⟫_ℝ • (v₁ j), ∑ j, ⟪v₁ j, v⟫_ℝ • (v₁ j)] --let t := ![∑ j, ⟪v₁ j, v⟫_ℝ • (v₁ j), ∑ j, ⟪v₁ j, v⟫_ℝ • (v₁ j)]
simp at t --simp at t
have L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F := by exact iteratedFDeriv ℝ 2 f z --have L : ContinuousMultilinearMap ℝ (fun (_ : Fin 2) ↦ E) F := by exact iteratedFDeriv ℝ 2 f z
--have α : Fin 2 → Type* := by exact fun _ ↦ ι --have α : Fin 2 → Type* := by exact fun _ ↦ ι
have g : (i : Fin 2) → ι → E := by exact fun _ ↦ (fun j ↦ ⟪v₁ j, v⟫_ℝ • (v₁ j)) --have g : (i : Fin 2) → ι → E := by exact fun _ ↦ (fun j ↦ ⟪v₁ j, v⟫_ℝ • (v₁ j))
have A : (i : Fin 2) → Finset ι := by exact fun _ ↦ Finset.univ --have A : (i : Fin 2) → Finset ι := by exact fun _ ↦ Finset.univ
let X := ContinuousMultilinearMap.map_sum let X := ContinuousMultilinearMap.map_sum
(iteratedFDeriv ℝ 2 f z) (iteratedFDeriv ℝ 2 f z)
@ -70,3 +66,16 @@ theorem LaplaceIndep
simp at X simp at X
sorry sorry
noncomputable def Laplace₁ (n : ℕ) (f : EuclideanSpace ℝ (Fin n) → F) : EuclideanSpace ℝ (Fin n) → F := by
let e : Fin n → EuclideanSpace ℝ (Fin n) := fun i ↦ EuclideanSpace.single i (1 : ℝ)
exact fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![e i, e i]
noncomputable def Laplace₂
[Fintype ι]
(v : Basis ι ℝ E)
(hv : Orthonormal ℝ v)
(f : E → F) :
E → F :=
fun z ↦ ∑ i, iteratedFDeriv ℝ 2 f z ![v i, v i]