Prove2Me

`BookProof.OperatorSeries.seriesOp_apply` (hnorm : ∀ (k : κ) (x : maxDom c), ‖(T k x : L2I ι)‖ ≤ a k * ‖(diagMax c x : L2I ι)‖) (ha : Summable a) (x : maxDom c) : (seriesOp T a hno

Proved
BookProof.OperatorSeries.seriesOp_apply

by leonardopedro · Oct 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterOperatorSeriesEsa.

BookProof.OperatorSeries.seriesOp_apply (hnorm : ∀ (k : κ) (x : maxDom c), ‖(T k x : L2I ι)‖ ≤ a k * ‖(diagMax c x : L2I ι)‖) (ha : Summable a) (x : maxDom c) : (seriesOp T a hnorm ha x : L2I ι) = ∑' k, (T k x : L2I ι)

Formalization note: Lean 4 identifier BookProof.OperatorSeries.seriesOp_apply.

Preamble
-- Generated from ChapterOperatorSeriesEsa.lean — theorem BookProof.OperatorSeries.seriesOp_apply
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterNavierStokesAffineFiberEsa
import Definitions.Def_ChapterNavierStokesDeficiency
import Mathlib
import Definitions.Def_ChapterOperatorSeriesEsa
import Definitions.Def_ChapterNavierStokesIkebeKato
open BookProof.NavierStokesFlow.IkebeKato
open BookProof.OperatorSeries

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] {D : Submodule ℂ F}
variable {ι κ : Type*} {c : ι → ℝ}
variable (T : κ → (maxDom c →ₗ[ℂ] L2I ι)) (a : κ → ℝ)
variable {T} {a}



open BookProof.FarisLavine BookProof.NavierStokesFlow BookProof.NavierStokesFlow.IkebeKato
open BookProof.NavierStokesFlow.LpNat

noncomputable section
Formal statement
theorem BookProof.OperatorSeries.seriesOp_apply
    (hnorm : ∀ (k : κ) (x : maxDom c), ‖(T k x : L2I ι)‖ ≤ a k * ‖(diagMax c x : L2I ι)‖)
    (ha : Summable a) (x : maxDom c) :
    (seriesOp T a hnorm ha x : L2I ι) = ∑' k, (T k x : L2I ι) := by sorry

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