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`BookProof.ChapterCyclicDirectSum.hasSum_starProjection_cyclicSubspace` {S : Set H} (hS : OrthogonalCyclicFamily T hT S) (htop : (⨆ x ∈ S, cyclicSubspace T hT x).topologicalClosure

Proved
BookProof.ChapterCyclicDirectSum.hasSum_starProjection_cyclicSubspace

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCyclicDirectSum.

BookProof.ChapterCyclicDirectSum.hasSum_starProjection_cyclicSubspace {S : Set H} (hS : OrthogonalCyclicFamily T hT S) (htop : (⨆ x ∈ S, cyclicSubspace T hT x).topologicalClosure = ⊤) (v : H) : HasSum (fun x : S => (cyclicSubspace T hT (x : H)).starProjection v) v

Formalization note: Lean 4 identifier BookProof.ChapterCyclicDirectSum.hasSum_starProjection_cyclicSubspace.

Preamble
-- Generated from ChapterCyclicDirectSum.lean — theorem BookProof.ChapterCyclicDirectSum.hasSum_starProjection_cyclicSubspace
import Definitions.Def_ChapterCyclicDecomposition
import Mathlib
import Definitions.Def_ChapterCyclicDirectSum
open BookProof.ChapterCyclicDirectSum


noncomputable section

open MeasureTheory Complex


open BookProof.ChapterCyclicDecomposition

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (T : H →L[ℂ] H) (hT : IsStarNormal T)
Formal statement
theorem BookProof.ChapterCyclicDirectSum.hasSum_starProjection_cyclicSubspace {S : Set H} (hS : OrthogonalCyclicFamily T hT S)
    (htop : (⨆ x ∈ S, cyclicSubspace T hT x).topologicalClosure = ⊤) (v : H) :
    HasSum (fun x : S => (cyclicSubspace T hT (x : H)).starProjection v) v := by sorry

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