`BookProof.ClosureUniqueness.eqOn_topologicalClosure_range_of_eqOn_range` (B : D →ₗ[ℂ] F) (U V : F →L[ℂ] F) (h : ∀ x : D, U (B x) = V (B x)) : ∀ z ∈ (LinearMap.range B).topological

Proved

BookProof.ClosureUniqueness.eqOn_topologicalClosure_range_of_eqOn_range

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterClosureUniqueness.

BookProof.ClosureUniqueness.eqOn_topologicalClosure_range_of_eqOn_range (B : D →ₗ[ℂ] F) (U V : F →L[ℂ] F) (h : ∀ x : D, U (B x) = V (B x)) : ∀ z ∈ (LinearMap.range B).topologicalClosure, U z = V z

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

Preamble

-- Generated from ChapterClosureUniqueness.lean — theorem BookProof.ClosureUniqueness.eqOn_topologicalClosure_range_of_eqOn_range
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterEsaClosureCore
import Mathlib
import Definitions.Def_ChapterClosureUniqueness
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.eqOn_topologicalClosure_range_of_eqOn_range (B : D →ₗ[ℂ] F) (U V : F →L[ℂ] F)
    (h : ∀ x : D, U (B x) = V (B x)) :
    ∀ z ∈ (LinearMap.range B).topologicalClosure, U z = V z := by sorry

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