Theorems · Theorem · functional analysis

DirectSum.IsInternal.collectedBasis_orthonormal

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_4} [inst_3 : DecidableEq ι] {V : ι → Submodule 𝕜 E},
  (OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ) →
    ∀ (hV_sum : DirectSum.IsInternal fun i => V i) {α : ι → Type u_6}
      {v_family : (i : ι) → Module.Basis (α i) 𝕜 ↥(V i)},
      (∀ (i : ι), Orthonormal 𝕜 ⇑(v_family i)) → Orthonormal 𝕜 ⇑(hV_sum.collectedBasis v_family)

Defined in

Mathlib.Analysis.InnerProductSpace.Subspace

Cited by

0 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceDecidableEq