`BookProof.SpectralGapStability.gapAt_of_tendsto` {A : ℕ → F →L[ℂ] F} {B : F →L[ℂ] F} {lam d : ℝ} (hA : ∀ m, GapAt (A m) lam d) (hconv : Tendsto (fun m => ‖A m - B‖) atTop...

Proved

BookProof.SpectralGapStability.gapAt_of_tendsto

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSpectralGapStability.

BookProof.SpectralGapStability.gapAt_of_tendsto {A : ℕ → F →L[ℂ] F} {B : F →L[ℂ] F} {lam d : ℝ} (hA : ∀ m, GapAt (A m) lam d) (hconv : Tendsto (fun m => ‖A m - B‖) atTop (𝓝 0)) : GapAt B lam d

Formalization note: Lean 4 identifier BookProof.SpectralGapStability.gapAt_of_tendsto.

Preamble

-- Generated from ChapterSpectralGapStability.lean — theorem BookProof.SpectralGapStability.gapAt_of_tendsto
import Mathlib
import Definitions.Def_ChapterSpectralGapStability
open BookProof.SpectralGapStability


noncomputable section


open scoped InnerProductSpace
open Filter Topology

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

Formal statement

theorem BookProof.SpectralGapStability.gapAt_of_tendsto {A : ℕ → F →L[ℂ] F} {B : F →L[ℂ] F} {lam d : ℝ}
    (hA : ∀ m, GapAt (A m) lam d)
    (hconv : Tendsto (fun m => ‖A m - B‖) atTop (𝓝 0)) : GapAt B lam d := by sorry

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