Theorems · Theorem · functional analysis

Submodule.starProjection_tendsto_closure_iSup

∀ {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 : Preorder ι] (U : ι → Submodule 𝕜 E) [inst_4 : ∀ (i : ι), (U i).HasOrthogonalProjection]
  [inst_5 : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection],
  Monotone U →
    ∀ (x : E),
      Filter.Tendsto (fun i => (U i).starProjection x) Filter.atTop
        (nhds ((⨆ i, U i).topologicalClosure.starProjection x))

Given a monotone family U of complete submodules of E and a fixed x : E, the orthogonal projection of x on U i tends to the orthogonal projection of x on (⨆ i, U i).topologicalClosure along atTop.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Cited by

1 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpacePreorderSubmodule.HasOrthogonalProjectionSubmodule.HasOrthogonalProjection