Theorems · Theorem · probability
ProbabilityTheory.covarianceBilin_map
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : MeasurableSpace E]
[inst_3 : BorelSpace E] {μ : MeasureTheory.Measure E} {F : Type u_2} [inst_4 : NormedAddCommGroup F]
[inst_5 : InnerProductSpace ℝ F] [inst_6 : MeasurableSpace F] [inst_7 : BorelSpace F] [SecondCountableTopology F]
[inst_9 : CompleteSpace F] [inst_10 : CompleteSpace E] [MeasureTheory.IsFiniteMeasure μ],
MeasureTheory.MemLp id 2 μ →
∀ (L : E →L[ℝ] F) (u v : F),
((ProbabilityTheory.covarianceBilin (MeasureTheory.Measure.map (⇑L) μ)) u) v =
((ProbabilityTheory.covarianceBilin μ) ((ContinuousLinearMap.adjoint L) u)) ((ContinuousLinearMap.adjoint L) v)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 274 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites35
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
- Realstatement and proof · cited by 25,697
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement · cited by 9,879
- ContinuousLinearMapstatement and proof · cited by 5,352
- InnerProductSpacestatement and proof · cited by 3,523
- CompleteSpacestatement and proof · cited by 2,532
- MeasureTheory.integralproof · cited by 1,779
- BorelSpacestatement and proof · cited by 1,602
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.covarianceBilin_multivariateGaussianproof · cited by 3