`BookProof.KatoRellich.dense_range_add_bounded` (H : D →ₗ[ℂ] F) (hH : SymmetricOn D H) (B : F →L[ℂ] F) (e : ℝ) (he : ‖B‖ < |e|) (hdense : Dense (Set.range fun x : D => H x - ((e :

Proved

BookProof.KatoRellich.dense_range_add_bounded

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterKatoRellichDeficiency.

BookProof.KatoRellich.dense_range_add_bounded (H : D →ₗ[ℂ] F) (hH : SymmetricOn D H) (B : F →L[ℂ] F) (e : ℝ) (he : ‖B‖ < |e|) (hdense : Dense (Set.range fun x : D => H x - ((e : ℂ) * Complex.I) • (x : F))) : Dense (Set.range fun x : D => (H x + B (x : F)) - ((e : ℂ) * Complex.I) • (x : F))

Formalization note: Lean 4 identifier BookProof.KatoRellich.dense_range_add_bounded.

Preamble

-- Generated from ChapterKatoRellichDeficiency.lean — theorem BookProof.KatoRellich.dense_range_add_bounded
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterKatoRellichDeficiency
import Definitions.Def_ChapterFarisLavineCore




open BookProof.FarisLavine

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] {D : Submodule ℂ F}

Formal statement

theorem BookProof.KatoRellich.dense_range_add_bounded (H : D →ₗ[ℂ] F) (hH : SymmetricOn D H)
    (B : F →L[ℂ] F) (e : ℝ) (he : ‖B‖ < |e|)
    (hdense : Dense (Set.range fun x : D => H x - ((e : ℂ) * Complex.I) • (x : F))) :
    Dense (Set.range fun x : D => (H x + B (x : F)) - ((e : ℂ) * Complex.I) • (x : F)) := by sorry

View graph