`BookProof.ChapterSolidHarmonic.sum_nullCLM_sq` {u v : E} (hu : ‖u‖ = 1) (hv : ‖v‖ = 1) (huv : ⟪u, v⟫_ℝ = 0) : ∑ i, (nullCLM u v (stdOrthonormalBasis ℝ E i)) ^ 2 = 0

Proved

BookProof.ChapterSolidHarmonic.sum_nullCLM_sq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSolidHarmonic.

BookProof.ChapterSolidHarmonic.sum_nullCLM_sq {u v : E} (hu : ‖u‖ = 1) (hv : ‖v‖ = 1) (huv : ⟪u, v⟫_ℝ = 0) : ∑ i, (nullCLM u v (stdOrthonormalBasis ℝ E i)) ^ 2 = 0

Formalization note: Lean 4 identifier BookProof.ChapterSolidHarmonic.sum_nullCLM_sq.

Preamble

-- Generated from ChapterSolidHarmonic.lean — theorem BookProof.ChapterSolidHarmonic.sum_nullCLM_sq
import Definitions.Def_ChapterLaplacianProduct
import Definitions.Def_ChapterSolidHarmonicTools
import Definitions.Def_ChapterLegendrePolynomial
import Mathlib
import Definitions.Def_ChapterSolidHarmonic
import Definitions.Def_ChapterRadialLaplacian
open BookProof.ChapterRadialLaplacian
open BookProof.ChapterSolidHarmonic



open Laplacian InnerProductSpace Polynomial
open BookProof.ChapterRadialLaplacian BookProof.ChapterLaplacianProduct
open BookProof.ChapterSolidHarmonicTools BookProof.ChapterLegendrePolynomial
open scoped RealInnerProductSpace

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

Formal statement

theorem BookProof.ChapterSolidHarmonic.sum_nullCLM_sq {u v : E} (hu : ‖u‖ = 1) (hv : ‖v‖ = 1) (huv : ⟪u, v⟫_ℝ = 0) :
    ∑ i, (nullCLM u v (stdOrthonormalBasis ℝ E i)) ^ 2 = 0 := by sorry

View graph