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Stefan Kebekus 2024-06-23 21:13:01 +02:00
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import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
open BigOperators
open Finset
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace E] [FiniteDimensional E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace F]
#check EuclideanSpace.norm_eq
#check EuclideanSpace.dist_eq
noncomputable def Laplace₁ (n : ) (f : EuclideanSpace (Fin n) → F) : EuclideanSpace (Fin n) → F := by
let e : Fin n → EuclideanSpace (Fin n) := fun i ↦ EuclideanSpace.single i (1 : )
exact fun z ↦ ∑ i, iteratedFDeriv 2 f z ![e i, e i]
noncomputable def Laplace₂
[Fintype ι]
(v : Basis ι E)
(hv : Orthonormal v)
(f : E → F) :
E → F :=
fun z ↦ ∑ i, iteratedFDeriv 2 f z ![v i, v i]
theorem LaplaceIndep
[Fintype ι]
(v₁ : Basis ι E)
(hv₁ : Orthonormal v₁)
(v₂ : Basis ι E)
(hv₂ : Orthonormal v₂)
(f : E → F) :
∑ i, iteratedFDeriv 2 f z ![v₁ i, v₁ i] = ∑ i, iteratedFDeriv 2 f z ![v₂ i, v₂ i] := by
have (v : E) : v = ∑ j, ⟪v₁ j, v⟫_ • (v₁ j) :=
sorry
conv =>
right
arg 2
intro i
rw [this (v₂ i)]
rw [this (v₂ i)]
conv =>
right
arg 2
intro i
rw [ContinuousMultilinearMap.map_sum_finset]
sorry