Theorems · Definition · functional analysis

Submodule.orthogonalDecomposition

{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 ≃ₗᵢ[𝕜] WithLp 2 (↥K × ↥Kᗮ)

If a subspace K of an inner product space E admits an orthogonal projection, then E is isometrically isomorphic to the L² product of K and Kᗮ.

Defined in

Mathlib.Analysis.InnerProductSpace.ProdL2

Cited by

14 results in Mathlib

Foundations

Depth 224 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection