Theorems · Theorem · functional analysis

TemperedDistribution.lineDeriv_eq_fourierMultiplierCLM

∀ {E : Type u_3} {F : Type u_4} [inst : NormedAddCommGroup E] [inst_1 : NormedAddCommGroup F]
  [inst_2 : InnerProductSpace ℝ E] [inst_3 : NormedSpace ℂ F] [inst_4 : FiniteDimensional ℝ E]
  [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E] (m : E) (f : TemperedDistribution E F),
  LineDeriv.lineDerivOp m f =
    (2 * ↑Real.pi * Complex.I) • (TemperedDistribution.fourierMultiplierCLM F fun x => ↑(inner ℝ x m)) f

Defined in: Mathlib.Analysis.Distribution.FourierMultiplier
Cited by: 2 results in Mathlib
Foundations: Depth 305 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup NormedAddCommGroup InnerProductSpace NormedSpace FiniteDimensional MeasurableSpace BorelSpace

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement · cited by 53,352
Realstatement and proof · cited by 25,697
RingHom.idstatement · cited by 18,349
NormedAddCommGroupstatement and proof · cited by 15,752
MeasurableSpacestatement and proof · cited by 13,106
NormedSpacestatement and proof · cited by 12,499
Set.Elemstatement · cited by 7,166
Set.ofPredstatement · cited by 6,101
Complexstatement and proof · cited by 5,565
ContinuousLinearMapstatement · cited by 5,352
InnerProductSpacestatement and proof · cited by 3,523

Cited by2

Results whose statement or proof uses this declaration.

TemperedDistribution.laplacian_eq_fourierMultiplierCLMproof · cited by 1
TemperedDistribution.besselPotential_neg_one_lineDerivOp_eqproof · cited by 1