`BookProof.GraphCore.deficiencyTrivialAt_of_bounded_dense` {D : Submodule ℂ F} (T : D →ₗ[ℂ] F) (hT : SymmetricOn D T) {C : ℝ} (hC0 : 0 ≤ C) (hC : ∀ x : D, ‖T x‖ ≤ C * ‖(x :...

Proved

BookProof.GraphCore.deficiencyTrivialAt_of_bounded_dense

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGraphCoreTransfer.

BookProof.GraphCore.deficiencyTrivialAt_of_bounded_dense {D : Submodule ℂ F} (T : D →ₗ[ℂ] F) (hT : SymmetricOn D T) {C : ℝ} (hC0 : 0 ≤ C) (hC : ∀ x : D, ‖T x‖ ≤ C * ‖(x : F)‖) (hdense : Dense (D : Set F)) {d : ℝ} (hd : d ≠ 0) : DeficiencyTrivialAt D T ((d : ℂ) * Complex.I)

Formalization note: Lean 4 identifier BookProof.GraphCore.deficiencyTrivialAt_of_bounded_dense.

Preamble

-- Generated from ChapterGraphCoreTransfer.lean — theorem BookProof.GraphCore.deficiencyTrivialAt_of_bounded_dense
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterGraphCoreTransfer
import Definitions.Def_ChapterComplexShiftCore
import Definitions.Def_ChapterFarisLavineCore
open BookProof.HashimotoShiftInvert
open BookProof.GraphCore



open BookProof.FarisLavine

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

Formal statement

theorem BookProof.GraphCore.deficiencyTrivialAt_of_bounded_dense {D : Submodule ℂ F} (T : D →ₗ[ℂ] F)
    (hT : SymmetricOn D T) {C : ℝ} (hC0 : 0 ≤ C) (hC : ∀ x : D, ‖T x‖ ≤ C * ‖(x : F)‖)
    (hdense : Dense (D : Set F)) {d : ℝ} (hd : d ≠ 0) :
    DeficiencyTrivialAt D T ((d : ℂ) * Complex.I) := by sorry

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