Theorems · Theorem · functional analysis
OrthonormalBasis.sum_repr
∀ {ι : Type u_1} {𝕜 : Type u_3} [inst : RCLike 𝕜] {E : Type u_4} [inst_1 : NormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] [inst_3 : Fintype ι] (b : OrthonormalBasis ι 𝕜 E) (x : E),
∑ i, (b.repr x).ofLp i • b i = x- Defined in
- Mathlib.Analysis.InnerProductSpace.PiL2
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 227 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- ENNRealstatement · cited by 9,879
- Fintypestatement and proof · cited by 7,736
- Finset.sumstatement · cited by 5,195
- InnerProductSpacestatement and proof · cited by 3,523
- Finset.univstatement and proof · cited by 3,473
- RCLikestatement and proof · cited by 2,829
- Finset.sum_congrproof · cited by 2,323
- LinearIsometryEquivstatement · cited by 748
- WithLp.ofLpstatement and proof · cited by 323
Cited by7
Results whose statement or proof uses this declaration.
- OrthonormalBasis.sum_inner_mul_innerproof · cited by 3
- OrthonormalBasis.sum_repr'proof · cited by 3
- OrthonormalBasis.orthogonalProjectionOnto_apply_eq_sumproof · cited by 2
- ContinuousLinearMap.isPositive_iff_eq_sum_rankOneproof · cited by 1
- OrthonormalBasis.equiv_applyproof · cited by 1
- ProbabilityTheory.HasGaussianLaw.indepFun_of_covariance_evalproof · cited by 0
- ProbabilityTheory.HasGaussianLaw.iIndepFun_of_covariance_evalproof · cited by 0