`BookProof.ChapterAbelianGelfandModel.inner_oneVec_mulRep` [IsFiniteMeasure mu] (f : C(X, ℂ)) : inner ℂ (oneVec mu) (mulRep mu f (oneVec mu)) = ∫ x, f x ∂mu

Proved

BookProof.ChapterAbelianGelfandModel.inner_oneVec_mulRep

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianGelfandModel.

BookProof.ChapterAbelianGelfandModel.inner_oneVec_mulRep [IsFiniteMeasure mu] (f : C(X, ℂ)) : inner ℂ (oneVec mu) (mulRep mu f (oneVec mu)) = ∫ x, f x ∂mu

Formalization note: Lean 4 identifier BookProof.ChapterAbelianGelfandModel.inner_oneVec_mulRep.

Preamble

-- Generated from ChapterAbelianGelfandModel.lean — theorem BookProof.ChapterAbelianGelfandModel.inner_oneVec_mulRep
import Definitions.Def_ChapterLinftyMultiplication
import Mathlib
import Definitions.Def_ChapterAbelianGelfandModel
open BookProof.ChapterAbelianGelfandModel


open MeasureTheory Complex WeakDual CompactlySupported CompactlySupportedContinuousMap
open scoped ComplexOrder


open BookProof.ChapterLinftyMultiplication

variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
variable {X : Type*} [TopologicalSpace X]
variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
  (psi : C(X, ℂ) →ₗ[ℂ] ℂ) (hpos : ∀ g : C(X, ℂ), 0 ≤ psi (star g * g))
variable {X : Type*} [MeasurableSpace X] (mu : Measure X)
variable {X : Type*} [TopologicalSpace X] [CompactSpace X]
  [MeasurableSpace X] [BorelSpace X] (mu : Measure X)

Formal statement

theorem BookProof.ChapterAbelianGelfandModel.inner_oneVec_mulRep [IsFiniteMeasure mu] (f : C(X, ℂ)) :
    inner ℂ (oneVec mu) (mulRep mu f (oneVec mu)) = ∫ x, f x ∂mu := by sorry

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