Prove2Me

`BookProof.ChapterCyclicDecomposition.exists_cyclic_decomposition` : ∃ S : Set H, OrthogonalCyclicFamily T hT S ∧ (⨆ x ∈ S, cyclicSubspace T hT x).topologicalClosure = ⊤

Proved
BookProof.ChapterCyclicDecomposition.exists_cyclic_decomposition

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCyclicDecomposition.

BookProof.ChapterCyclicDecomposition.exists_cyclic_decomposition : ∃ S : Set H, OrthogonalCyclicFamily T hT S ∧ (⨆ x ∈ S, cyclicSubspace T hT x).topologicalClosure = ⊤

Formalization note: Lean 4 identifier BookProof.ChapterCyclicDecomposition.exists_cyclic_decomposition.

Preamble
-- Generated from ChapterCyclicDecomposition.lean — theorem BookProof.ChapterCyclicDecomposition.exists_cyclic_decomposition
import Definitions.Def_ChapterSpectralMultiplication
import Mathlib
import Definitions.Def_ChapterCyclicDecomposition
open BookProof.ChapterCyclicDecomposition


noncomputable section

open MeasureTheory Complex


open BookProof.ChapterSpectralMultiplication

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (T : H →L[ℂ] H) (hT : IsStarNormal T)
Formal statement
theorem BookProof.ChapterCyclicDecomposition.exists_cyclic_decomposition :
    ∃ S : Set H, OrthogonalCyclicFamily T hT S ∧
      (⨆ x ∈ S, cyclicSubspace T hT x).topologicalClosure = ⊤ := by sorry

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