Theorems · Definition · functional analysis

Submodule.quotientEquivOrthogonal

{𝕜 : Type u_1} →
  {E : Type u_4} →
    [inst : RCLike 𝕜] →
      [inst_1 : NormedAddCommGroup E] →
        [inst_2 : InnerProductSpace 𝕜 E] → (K : Submodule 𝕜 E) → [K.HasOrthogonalProjection] → E ⧸ K ≃ₗᵢ[𝕜] ↥Kᗮ

If a subspace K of an inner product space E admits an orthogonal projection, then the quotient E ⧸ K is isometrically isomorphic to the orthogonal complement Kᗮ of K.

Defined in

Mathlib.Analysis.InnerProductSpace.ProdL2

Cited by

7 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection