Theorems · Theorem · functional analysis

OrthogonalFamily.projection_directSum_coeAddHom

∀ {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) →
    ∀ (x : DirectSum ι fun i => ↥(V i)) (i : ι) [inst_4 : CompleteSpace ↥(V i)],
      (V i).orthogonalProjectionOnto ((DirectSum.coeAddMonoidHom V) x) = x i

If a family of submodules is orthogonal, then the orthogonalProjection on a direct sum is just the coefficient of that direct sum.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Cited by

0 results in Mathlib

Foundations

Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceDecidableEqCompleteSpace