Theorems · Theorem · measure theory

OrthonormalBasis.volume_parallelepiped

∀ {ι : Type u_1} {F : Type u_3} [inst : NormedAddCommGroup F] [inst_1 : InnerProductSpace ℝ F]
  [inst_2 : MeasurableSpace F] [inst_3 : BorelSpace F] [inst_4 : Fintype ι] [inst_5 : FiniteDimensional ℝ F]
  (b : OrthonormalBasis ι ℝ F), MeasureTheory.volume (parallelepiped ⇑b) = 1

The volume measure in a finite-dimensional inner product space gives measure 1 to the parallelepiped spanned by any orthonormal basis.

Defined in: Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
Cited by: 4 results in Mathlib
Foundations: Depth 263 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup InnerProductSpace MeasurableSpace BorelSpace Fintype FiniteDimensional

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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
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
Fintypestatement and proof · cited by 7,736
InnerProductSpacestatement and proof · cited by 3,523
Factproof · cited by 2,726
FiniteDimensionalstatement and proof · cited by 1,854
Module.finrankproof · cited by 1,770

Cited by4

Results whose statement or proof uses this declaration.

OrthonormalBasis.addHaar_eq_volumeproof · cited by 4
Complex.volume_preserving_equiv_piproof · cited by 2
MeasureTheory.Measure.euclideanHausdorffMeasure_zeroproof · cited by 0
MeasureTheory.Measure.addHaarScalarFactor_volume_hausdorffMeasure_ne_zeroproof · cited by 0