Theorems · Theorem · functional analysis

Submodule.sup_orthogonal_inf_of_hasOrthogonalProjection

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {K₁ K₂ : Submodule 𝕜 E}, K₁ ≤ K₂ → ∀ [K₁.HasOrthogonalProjection], K₁ ⊔ K₁ᗮ ⊓ K₂ = K₂

If K₁ admits an orthogonal projection and is contained in K₂, then K₁ and K₁ᗮ ⊓ K₂ span K₂.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Cited by

2 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection