`BookProof.ChapterTensorCompleteness.hasSum_tensorHilbertBasis` (b : HilbertBasis ι ℂ (Lp ℂ 2 μ)) (c : HilbertBasis κ ℂ (Lp ℂ 2 ν)) (F : Lp ℂ 2 (μ.prod ν)) : HasSum (fun p : ι × κ

Proved

BookProof.ChapterTensorCompleteness.hasSum_tensorHilbertBasis

by leonardopedro · Oct 22, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTensorCompleteness.

BookProof.ChapterTensorCompleteness.hasSum_tensorHilbertBasis (b : HilbertBasis ι ℂ (Lp ℂ 2 μ)) (c : HilbertBasis κ ℂ (Lp ℂ 2 ν)) (F : Lp ℂ 2 (μ.prod ν)) : HasSum (fun p : ι × κ => (inner ℂ (tensorOf (b p.1) (c p.2)) F : ℂ) • tensorOf (b p.1) (c p.2)) F

Formalization note: Lean 4 identifier BookProof.ChapterTensorCompleteness.hasSum_tensorHilbertBasis.

Preamble

-- Generated from ChapterTensorCompleteness.lean — theorem BookProof.ChapterTensorCompleteness.hasSum_tensorHilbertBasis
import Definitions.Def_ChapterSolovayHilbertTensor
import Mathlib
import Definitions.Def_ChapterTensorCompleteness
open BookProof.ChapterTensorCompleteness


noncomputable section

open MeasureTheory ENNReal Complex Filter Topology


open BookProof.ChapterSolovayHilbertTensor

variable {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
  {μ : Measure α} {ν : Measure β} [IsFiniteMeasure μ] [IsFiniteMeasure ν]

variable {ι κ : Type*}

Formal statement

theorem BookProof.ChapterTensorCompleteness.hasSum_tensorHilbertBasis
    (b : HilbertBasis ι ℂ (Lp ℂ 2 μ)) (c : HilbertBasis κ ℂ (Lp ℂ 2 ν))
    (F : Lp ℂ 2 (μ.prod ν)) :
    HasSum (fun p : ι × κ =>
        (inner ℂ (tensorOf (b p.1) (c p.2)) F : ℂ) • tensorOf (b p.1) (c p.2)) F := by sorry

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