Theorems · Definition · functional analysis

DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv

{ι : Type u_7} →
  {𝕜 : Type u_8} →
    [inst : RCLike 𝕜] →
      {E : Type u_9} →
        [inst_1 : NormedAddCommGroup E] →
          [inst_2 : InnerProductSpace 𝕜 E] →
            [Fintype ι] →
              [FiniteDimensional 𝕜 E] →
                {n : ℕ} →
                  Module.finrank 𝕜 E = n →
                    [inst_5 : DecidableEq ι] →
                      {V : ι → Submodule 𝕜 E} →
                        DirectSum.IsInternal V →
                          (OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ) →
                            (i : ι) × Fin (Module.finrank 𝕜 ↥(V i)) ≃ Fin n

Exhibit a bijection between Fin n and the index set of a certain basis of an n-dimensional inner product space E. This should not be accessed directly, but only via the subsequent API.

Defined in

Mathlib.Analysis.InnerProductSpace.PiL2

Cited by

6 results in Mathlib

Foundations

Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFintypeFiniteDimensionalDecidableEq