Theorems · Theorem · functional analysis
OrthonormalBasis.sum_sq_inner_left
∀ {ι : Type u_7} {E : Type u_8} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : Fintype ι]
(b : OrthonormalBasis ι ℝ E) (x : E), ∑ i, inner ℝ x (b i) ^ 2 = ‖x‖ ^ 2- Defined in
- Mathlib.Analysis.InnerProductSpace.PiL2
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 233 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- NormedAddCommGroupstatement and proof · cited by 15,752
- Fintypestatement and proof · cited by 7,736
- Norm.normstatement · cited by 5,413
- Finset.sumstatement and proof · cited by 5,195
- InnerProductSpacestatement and proof · cited by 3,523
- Finset.univstatement and proof · cited by 3,473
- Finset.sum_congrproof · cited by 2,323
- Inner.innerstatement and proof · cited by 1,089
- OrthonormalBasisstatement and proof · cited by 188
- real_inner_commproof · cited by 57
Cited by3
Results whose statement or proof uses this declaration.
- TemperedDistribution.laplacian_eq_fourierMultiplierCLMproof · cited by 1
- OrthonormalBasis.norm_dualproof · cited by 1
- SchwartzMap.laplacian_eq_fourierMultiplierCLMproof · cited by 0