`BookProof.ChapterMackeyImprimitivity.ImprimitivitySystem.pvm_parseval` (ψ : E) : ∑ x : X, ‖S.p x ψ‖ ^ 2 = ‖ψ‖ ^ 2

Proved

BookProof.ChapterMackeyImprimitivity.ImprimitivitySystem.pvm_parseval

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMackeyImprimitivity.

BookProof.ChapterMackeyImprimitivity.ImprimitivitySystem.pvm_parseval (ψ : E) : ∑ x : X, ‖S.p x ψ‖ ^ 2 = ‖ψ‖ ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterMackeyImprimitivity.ImprimitivitySystem.pvm_parseval.

Preamble

-- Generated from ChapterMackeyImprimitivity.lean — theorem BookProof.ChapterMackeyImprimitivity.ImprimitivitySystem.pvm_parseval
import Mathlib
import Definitions.Def_ChapterMackeyImprimitivity
import Definitions.Def_ChapterStoneConverse
open BookProof.ChapterStoneMeasurable
open BookProof.ChapterStoneMeasurable.WeakMeasurableUnitaryGroup
open BookProof.ChapterMackeyImprimitivity
open BookProof.ChapterMackeyImprimitivity


open scoped InnerProductSpace
open Finset


variable {G : Type*} [Group G] {X : Type*} [Fintype X] [MulAction G X]
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]

variable (S : ImprimitivitySystem G X E)

Formal statement

theorem BookProof.ChapterMackeyImprimitivity.ImprimitivitySystem.pvm_parseval (ψ : E) : ∑ x : X, ‖S.p x ψ‖ ^ 2 = ‖ψ‖ ^ 2 := by sorry

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