Theorems · Definition · functional analysis

OrthonormalBasis.mk

{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] →
              {v : ι → E} → Orthonormal 𝕜 v → ⊤ ≤ Submodule.span 𝕜 (Set.range v) → OrthonormalBasis ι 𝕜 E

A finite orthonormal set that spans is an orthonormal basis

Defined in: Mathlib.Analysis.InnerProductSpace.PiL2
Cited by: 2 results in Mathlib
Foundations: Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: RCLike NormedAddCommGroup InnerProductSpace Fintype

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites12

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

NormedAddCommGroupstatement and proof · cited by 15,752
Top.topstatement and proof · cited by 9,680
Fintypestatement and proof · cited by 7,736
Submodulestatement · cited by 7,192
Set.rangestatement and proof · cited by 4,705
InnerProductSpacestatement and proof · cited by 3,523
RCLikestatement and proof · cited by 2,829
Submodule.spanstatement and proof · cited by 1,504
OrthonormalBasisstatement · cited by 188
Orthonormalstatement and proof · cited by 85
Module.Basis.mkproof · cited by 29
Module.Basis.toOrthonormalBasisproof · cited by 5

Cited by5

Results whose statement or proof uses this declaration.

OrthonormalBasis.coe_mkstatement · cited by 3
OrthonormalBasis.mkOfOrthogonalEqBotproof · cited by 3
HilbertBasis.toOrthonormalBasisproof · cited by 1
OrthonormalBasis.spanproof · cited by 1
OrthonormalBasis.mk.congr_simpstatement and proof · cited by 0