`BookProof.ChapterSolidHarmonicTools.sum_inner_sq` (e : E) : ∑ i, (⟪e, (stdOrthonormalBasis ℝ E) i⟫_ℝ) ^ 2 = ‖e‖ ^ 2

Proved

BookProof.ChapterSolidHarmonicTools.sum_inner_sq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonicTools.

BookProof.ChapterSolidHarmonicTools.sum_inner_sq (e : E) : ∑ i, (⟪e, (stdOrthonormalBasis ℝ E) i⟫_ℝ) ^ 2 = ‖e‖ ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonicTools.sum_inner_sq.

Preamble

-- Generated from ChapterSolidHarmonicTools.lean — theorem BookProof.ChapterSolidHarmonicTools.sum_inner_sq
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterLaplacianProduct
import Mathlib
import Definitions.Def_ChapterSolidHarmonicTools
open BookProof.ChapterSolidHarmonicTools



open Laplacian InnerProductSpace BookProof.ChapterRadialLaplacian
open BookProof.ChapterLaplacianProduct
open scoped RealInnerProductSpace


variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]

Formal statement

theorem BookProof.ChapterSolidHarmonicTools.sum_inner_sq (e : E) :
    ∑ i, (⟪e, (stdOrthonormalBasis ℝ E) i⟫_ℝ) ^ 2 = ‖e‖ ^ 2 := by sorry

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