`BookProof.ChapterAbelianCyclicModel.mulRep_toLp` (g f : C(X, ℂ)) : mulRep (repMeasure pi xi) g (ContinuousMap.toLp 2 (repMeasure pi xi) ℂ f) = ContinuousMap.toLp 2 (repMeasure pi

Proved

BookProof.ChapterAbelianCyclicModel.mulRep_toLp

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianCyclicModel.

BookProof.ChapterAbelianCyclicModel.mulRep_toLp (g f : C(X, ℂ)) : mulRep (repMeasure pi xi) g (ContinuousMap.toLp 2 (repMeasure pi xi) ℂ f) = ContinuousMap.toLp 2 (repMeasure pi xi) ℂ (g * f)

Formalization note: Lean 4 identifier BookProof.ChapterAbelianCyclicModel.mulRep_toLp.

Preamble

-- Generated from ChapterAbelianCyclicModel.lean — theorem BookProof.ChapterAbelianCyclicModel.mulRep_toLp
import Definitions.Def_ChapterAbelianGelfandModel
import Mathlib
import Definitions.Def_ChapterAbelianCyclicModel
open BookProof.ChapterAbelianCyclicModel


open MeasureTheory Complex WeakDual
open scoped ComplexOrder


open BookProof.ChapterAbelianGelfandModel


variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (pi : C(X, ℂ) →⋆ₐ[ℂ] (H →L[ℂ] H)) (xi : H)

variable (hcyc : DenseRange (repVec pi xi))

Formal statement

theorem BookProof.ChapterAbelianCyclicModel.mulRep_toLp (g f : C(X, ℂ)) :
    mulRep (repMeasure pi xi) g (ContinuousMap.toLp 2 (repMeasure pi xi) ℂ f)
      = ContinuousMap.toLp 2 (repMeasure pi xi) ℂ (g * f) := by sorry

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