`BookProof.ChapterCyclicDirectSum.exists_isHilbertSum_cyclicSubspace` : ∃ S : Set H, OrthogonalCyclicFamily T hT S ∧ IsHilbertSum ℂ (fun x : S => (cyclicSubspace T hT (x : H))) (fu

Proved

BookProof.ChapterCyclicDirectSum.exists_isHilbertSum_cyclicSubspace

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCyclicDirectSum.

BookProof.ChapterCyclicDirectSum.exists_isHilbertSum_cyclicSubspace : ∃ S : Set H, OrthogonalCyclicFamily T hT S ∧ IsHilbertSum ℂ (fun x : S => (cyclicSubspace T hT (x : H))) (fun x : S => (cyclicSubspace T hT (x : H)).subtypeₗᵢ)

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

Preamble

-- Generated from ChapterCyclicDirectSum.lean — theorem BookProof.ChapterCyclicDirectSum.exists_isHilbertSum_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.exists_isHilbertSum_cyclicSubspace :
    ∃ S : Set H, OrthogonalCyclicFamily T hT S ∧
      IsHilbertSum ℂ (fun x : S => (cyclicSubspace T hT (x : H)))
        (fun x : S => (cyclicSubspace T hT (x : H)).subtypeₗᵢ) := by sorry

View graph