Theorems · Theorem · functional analysis

LineDeriv.laplacianCLM_eq_sum

∀ {ι : Type u_1} {E : Type u_6} {V₁ : Type u_8} {V₂ : Type u_9} {V₃ : Type u_10} [inst : LineDeriv E V₁ V₂]
  [inst_1 : LineDeriv E V₂ V₃] [inst_2 : AddCommGroup V₁] [inst_3 : AddCommGroup V₂] [inst_4 : AddCommGroup V₃]
  [inst_5 : NormedAddCommGroup E] [inst_6 : InnerProductSpace ℝ E] [inst_7 : FiniteDimensional ℝ E]
  [inst_8 : Module ℝ V₁] [inst_9 : Module ℝ V₂] [inst_10 : Module ℝ V₃] [inst_11 : TopologicalSpace V₁]
  [inst_12 : TopologicalSpace V₂] [inst_13 : TopologicalSpace V₃] [inst_14 : IsTopologicalAddGroup V₃]
  [inst_15 : LineDerivAdd E V₁ V₂] [inst_16 : LineDerivSMul ℝ E V₁ V₂] [inst_17 : ContinuousLineDeriv E V₁ V₂]
  [inst_18 : LineDerivAdd E V₂ V₃] [inst_19 : LineDerivSMul ℝ E V₂ V₃] [inst_20 : ContinuousLineDeriv E V₂ V₃]
  [LineDerivLeftSMul ℝ E V₁ V₂] [LineDerivLeftSMul ℝ E V₂ V₃] [inst_23 : Fintype ι] (v : OrthonormalBasis ι ℝ E)
  (f : V₁), (LineDeriv.laplacianCLM ℝ E V₁) f = ∑ i, LineDeriv.lineDerivOp (v i) (LineDeriv.lineDerivOp (v i) f)

Defined in

Mathlib.Analysis.Distribution.DerivNotation

Cited by

2 results in Mathlib

Foundations

Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LineDerivLineDerivAddCommGroupAddCommGroupAddCommGroupNormedAddCommGroupInnerProductSpaceFiniteDimensionalModuleModuleModuleTopologicalSpaceTopologicalSpaceTopologicalSpaceIsTopologicalAddGroupLineDerivAddLineDerivSMulContinuousLineDerivLineDerivAddLineDerivSMulContinuousLineDerivLineDerivLeftSMulLineDerivLeftSMulFintype