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`BookProof.ClosureUniqueness.not_isSelfAdjointExtension_clExt_of_deficiency` (T : D →ₗ[ℂ] F) (hdense : Dense (D : Set F)) (hsym : SymmetricOn D T) {w : F} (hw0 : w ≠ 0) (hw : ∀ v :

Proved
BookProof.ClosureUniqueness.not_isSelfAdjointExtension_clExt_of_deficiency

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterClosureUniqueness.

BookProof.ClosureUniqueness.not_isSelfAdjointExtension_clExt_of_deficiency (T : D →ₗ[ℂ] F) (hdense : Dense (D : Set F)) (hsym : SymmetricOn D T) {w : F} (hw0 : w ≠ 0) (hw : ∀ v : D, (inner ℂ (T v) w : ℂ) = Complex.I * inner ℂ (v : F) w) : ¬ IsSelfAdjointExtension T (clExt T hdense hsym)

Formalization note: Lean 4 identifier BookProof.ClosureUniqueness.not_isSelfAdjointExtension_clExt_of_deficiency.

Preamble
-- Generated from ChapterClosureUniqueness.lean — theorem BookProof.ClosureUniqueness.not_isSelfAdjointExtension_clExt_of_deficiency
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterClosureUniqueness
import Definitions.Def_ChapterEsaClosureCore
import Definitions.Def_ChapterFarisLavineCore
open BookProof.EsaClosure
open BookProof.ClosureUniqueness



open BookProof.FarisLavine BookProof.EsaClosure

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]
variable {D D₁ D₂ Dom Dom₁ Dom₂ : Submodule ℂ F}
Formal statement
theorem BookProof.ClosureUniqueness.not_isSelfAdjointExtension_clExt_of_deficiency (T : D →ₗ[ℂ] F)
    (hdense : Dense (D : Set F)) (hsym : SymmetricOn D T) {w : F} (hw0 : w ≠ 0)
    (hw : ∀ v : D, (inner ℂ (T v) w : ℂ) = Complex.I * inner ℂ (v : F) w) :
    ¬ IsSelfAdjointExtension T (clExt T hdense hsym) := by sorry

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