Theorems · Theorem · functional analysis
SchwartzMap.laplacian_eq_sum
∀ {ι : Type u_1} {E : Type u_5} {F : Type u_8} [inst : NormedAddCommGroup E] [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace ℝ F] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E] [inst_5 : Fintype ι]
(b : OrthonormalBasis ι ℝ E) (f : SchwartzMap E F),
Laplacian.laplacian f = ∑ i, LineDeriv.lineDerivOp (b i) (LineDeriv.lineDerivOp (b i) f)- Cited by
- 5 results in Mathlib
- Foundations
- Depth 246 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Fintypestatement and proof · cited by 7,736
- Finset.sumstatement · cited by 5,195
- InnerProductSpacestatement and proof · cited by 3,523
- Finset.univstatement · cited by 3,473
- FiniteDimensionalstatement and proof · cited by 1,854
- SchwartzMapstatement and proof · cited by 251
- OrthonormalBasisstatement and proof · cited by 188
- LineDeriv.lineDerivOpstatement · cited by 60
Cited by5
Results whose statement or proof uses this declaration.
- SchwartzMap.integral_bilinear_laplacian_right_eq_leftproof · cited by 3
- TemperedDistribution.laplacian_apply_applyproof · cited by 1
- SchwartzMap.laplacianCLM_eqproof · cited by 0
- SchwartzMap.laplacian_applyproof · cited by 0
- SchwartzMap.laplacian_eq_fourierMultiplierCLMproof · cited by 0