Theorems · Theorem · functional analysis
Orthonormal.linearIndependent
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{ι : Type u_4} {v : ι → E}, Orthonormal 𝕜 v → LinearIndependent 𝕜 vAn orthonormal set is linearly independent.
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 171 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.coeproof · cited by 62,936
- Finsuppproof · cited by 5,255
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Inner.innerproof · cited by 1,089
- LinearIndependentstatement · cited by 560
- Finsupp.extproof · cited by 399
- Finsupp.linearCombinationproof · cited by 269
- Orthonormalstatement and proof · cited by 85
- inner_zero_rightproof · cited by 40
- linearIndependent_iffproof · cited by 12
Cited by6
Results whose statement or proof uses this declaration.
- Orthonormal.exists_orthonormalBasis_extensionproof · cited by 2
- Orthonormal.exists_orthonormalBasis_extension_of_card_eqproof · cited by 1
- basisOfOrthonormalOfCardEqFinrankproof · cited by 1
- maximal_orthonormal_iff_basis_of_finiteDimensionalproof · cited by 0
- coe_basisOfOrthonormalOfCardEqFinrankproof · cited by 0
- Orthonormal.toSubtypeRangeproof · cited by 0