Theorems · Theorem · functional analysis

OrthogonalFamily.isInternal_iff_of_isComplete

∀ {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) →
    IsComplete ↑(iSup V) → (DirectSum.IsInternal V ↔ (iSup V)ᗮ = ⊥)

An orthogonal family of subspaces of E satisfies DirectSum.IsInternal (that is, they provide an internal direct sum decomposition of E) if and only if their span has trivial orthogonal complement.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Cited by

1 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceDecidableEq