Theorems · Theorem · measure theory

InnerProductSpace.volume_ball_of_dim_even

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E]
  [inst_3 : MeasurableSpace E] [inst_4 : BorelSpace E] [Nontrivial E] {k : ℕ},
  Module.finrank ℝ E = 2 * k →
    ∀ (x : E) (r : ℝ),
      MeasureTheory.volume (Metric.ball x r) =
        ENNReal.ofReal r ^ Module.finrank ℝ E * ENNReal.ofReal (Real.pi ^ k / ↑k.factorial)

Defined in: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
Cited by: 3 results in Mathlib
Foundations: Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup InnerProductSpace FiniteDimensional MeasurableSpace BorelSpace Nontrivial

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Cites26

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

DFunLike.coestatement · 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 · cited by 10,939
ENNRealstatement and proof · cited by 9,879
InnerProductSpacestatement and proof · cited by 3,523
Nontrivialstatement and proof · cited by 2,416
FiniteDimensionalstatement and proof · cited by 1,854
Real.pistatement and proof · cited by 1,774
Module.finrankstatement and proof · cited by 1,770

Cited by3

Results whose statement or proof uses this declaration.

Complex.volume_ballproof · cited by 3
EuclideanSpace.volume_ball_fin_twoproof · cited by 1
InnerProductSpace.volume_closedBall_of_dim_evenproof · cited by 0