`BookProof.ChapterLinftyMultiplication.multOp_inner_adjoint` (φ : α → ℂ) (hφ : MemLp φ ⊤ μ) (f g : Lp ℂ 2 μ) : inner ℂ (multOp φ hφ f) g = inner ℂ f (multOp (fun x => (starRingEnd

Proved

BookProof.ChapterLinftyMultiplication.multOp_inner_adjoint

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLinftyMultiplication.

BookProof.ChapterLinftyMultiplication.multOp_inner_adjoint (φ : α → ℂ) (hφ : MemLp φ ⊤ μ) (f g : Lp ℂ 2 μ) : inner ℂ (multOp φ hφ f) g = inner ℂ f (multOp (fun x => (starRingEnd ℂ) (φ x)) (memLp_top_conj hφ) g)

Formalization note: Lean 4 identifier BookProof.ChapterLinftyMultiplication.multOp_inner_adjoint.

Preamble

-- Generated from ChapterLinftyMultiplication.lean — theorem BookProof.ChapterLinftyMultiplication.multOp_inner_adjoint
import Mathlib
import Definitions.Def_ChapterLinftyMultiplication
import Theorems.Thm_BookProof_ChapterLinftyMultiplication_memLp_top_conj
open BookProof.ChapterLinftyMultiplication


noncomputable section

open MeasureTheory ENNReal Complex


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

Formal statement

theorem BookProof.ChapterLinftyMultiplication.multOp_inner_adjoint (φ : α → ℂ) (hφ : MemLp φ ⊤ μ) (f g : Lp ℂ 2 μ) :
    inner ℂ (multOp φ hφ f) g
      = inner ℂ f (multOp (fun x => (starRingEnd ℂ) (φ x)) (memLp_top_conj hφ) g) := by sorry

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