Prove2Me

`BookProof.ChapterLaplacianProduct.sum_inner_mul_apply` [FiniteDimensional ℝ E] (L : E →L[ℝ] ℝ) (x : E) : ∑ i, ⟪x, (stdOrthonormalBasis ℝ E) i⟫ * L ((stdOrthonormalBasis ℝ E) i) =

Proved
BookProof.ChapterLaplacianProduct.sum_inner_mul_apply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLaplacianProduct.

BookProof.ChapterLaplacianProduct.sum_inner_mul_apply [FiniteDimensional ℝ E] (L : E →L[ℝ] ℝ) (x : E) : ∑ i, ⟪x, (stdOrthonormalBasis ℝ E) i⟫ * L ((stdOrthonormalBasis ℝ E) i) = L x

Formalization note: Lean 4 identifier BookProof.ChapterLaplacianProduct.sum_inner_mul_apply.

Preamble
-- Generated from ChapterLaplacianProduct.lean — theorem BookProof.ChapterLaplacianProduct.sum_inner_mul_apply
import Definitions.Def_ChapterRadialLaplacian
import Mathlib
import Definitions.Def_ChapterLaplacianProduct
open BookProof.ChapterLaplacianProduct



open Filter Laplacian InnerProductSpace BookProof.ChapterRadialLaplacian
open scoped InnerProductSpace RealInnerProductSpace

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
Formal statement
theorem BookProof.ChapterLaplacianProduct.sum_inner_mul_apply [FiniteDimensional ℝ E] (L : E →L[ℝ] ℝ) (x : E) :
    ∑ i, ⟪x, (stdOrthonormalBasis ℝ E) i⟫ * L ((stdOrthonormalBasis ℝ E) i) = L x := by sorry

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