Theorems · Theorem · functional analysis

Orthonormal.isHilbertSum

∀ {ι : Type u_1} {𝕜 : Type u_2} [inst : RCLike 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : CompleteSpace E] {v : ι → E} (hv : Orthonormal 𝕜 v),
  ⊤ ≤ (Submodule.span 𝕜 (Set.range v)).topologicalClosure →
    IsHilbertSum 𝕜 (fun x => 𝕜) fun i => LinearIsometry.toSpanSingleton 𝕜 E ⋯

Given a total orthonormal family v : ι → E, E is a Hilbert sum of fun i : ι => 𝕜 relative to the family of linear isometries fun i k => k • v i.

Defined in

Mathlib.Analysis.InnerProductSpace.l2Space

Cited by

1 results in Mathlib

Foundations

Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace