Theorems · Definition · functional analysis

stdOrthonormalBasis

(𝕜 : Type u_7) →
  [inst : RCLike 𝕜] →
    (E : Type u_8) →
      [inst_1 : NormedAddCommGroup E] →
        [inst_2 : InnerProductSpace 𝕜 E] → [FiniteDimensional 𝕜 E] → OrthonormalBasis (Fin (Module.finrank 𝕜 E)) 𝕜 E

A finite-dimensional InnerProductSpace has an orthonormal basis.

Defined in: Mathlib.Analysis.InnerProductSpace.PiL2
Cited by: 60 results in Mathlib
Foundations: Depth 240 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: RCLike NormedAddCommGroup InnerProductSpace FiniteDimensional

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites6

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

NormedAddCommGroupstatement · cited by 15,752
InnerProductSpacestatement · cited by 3,523
RCLikestatement · cited by 2,829
FiniteDimensionalstatement · cited by 1,854
Module.finrankstatement · cited by 1,770
OrthonormalBasisstatement · cited by 188

Cited by67

Results whose statement or proof uses this declaration.

LinearMap.normDetproof · cited by 27
ProbabilityTheory.stdGaussianproof · cited by 14
InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasisstatement and proof · cited by 8
LinearMap.normDet_eq_norm_det_toMatrix_rangeRestrictproof · cited by 8
Orientation.finOrthonormalBasisproof · cited by 8
OrthonormalBasis.fromOrthogonalSpanSingletonproof · cited by 8
InnerProductSpace.laplacianWithin_eq_iteratedFDerivWithin_stdOrthonormalBasisstatement and proof · cited by 7
LinearMap.normDet_ne_zero_tfaeproof · cited by 6
LinearMap.normDet_eq_zero_iff_ker_ne_botproof · cited by 5
InnerProductSpace.canonicalCovariantTensorproof · cited by 5
OrthonormalBasis.volume_parallelepipedproof · cited by 4
Orientation.finOrthonormalBasis_orientationproof · cited by 4