Theorems · Theorem · functional analysis

Submodule.starProjection_tendsto_self

∀ {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 : ∀ (t : ι), (U t).HasOrthogonalProjection],
  Monotone U →
    ∀ (x : E),
      ⊤ ≤ (⨆ t, U t).topologicalClosure → Filter.Tendsto (fun t => (U t).starProjection x) Filter.atTop (nhds x)

Given a monotone family U of complete submodules of E with dense span supremum, and a fixed x : E, the orthogonal projection of x on U i tends to x along at_top.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Cited by

0 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpacePreorderSubmodule.HasOrthogonalProjection