`BookProof.ChapterWignerSymmetry.inner_sum_smul` (b : OrthonormalBasis ι ℂ E) (v w : ι → ℂ) : ⟪∑ j, v j • b j, ∑ j, w j • b j⟫_ℂ = ∑ k, conj (v k) * w k

Proved

BookProof.ChapterWignerSymmetry.inner_sum_smul

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWignerSymmetry.

BookProof.ChapterWignerSymmetry.inner_sum_smul (b : OrthonormalBasis ι ℂ E) (v w : ι → ℂ) : ⟪∑ j, v j • b j, ∑ j, w j • b j⟫_ℂ = ∑ k, conj (v k) * w k

Formalization note: Lean 4 identifier BookProof.ChapterWignerSymmetry.inner_sum_smul.

Preamble

-- Generated from ChapterWignerSymmetry.lean — theorem BookProof.ChapterWignerSymmetry.inner_sum_smul
import Mathlib
import Definitions.Def_ChapterWignerSymmetry
open BookProof.ChapterWignerSymmetry


open scoped InnerProductSpace ComplexConjugate
open Finset


variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]


variable {T : E → E}

variable {ι : Type*} [Fintype ι] [DecidableEq ι]
variable {S : (ι → ℂ) → (ι → ℂ)} {o : ι}

Formal statement

theorem BookProof.ChapterWignerSymmetry.inner_sum_smul (b : OrthonormalBasis ι ℂ E) (v w : ι → ℂ) :
    ⟪∑ j, v j • b j, ∑ j, w j • b j⟫_ℂ = ∑ k, conj (v k) * w k := by sorry

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