`BookProof.ChapterOrthogonalSums.hasSum_smul_of_hasSum_norm_sq` [CompleteSpace E] {ι : Type*} {v : ι → E} (hv : Orthonormal ℂ v) {y : E} (h : HasSum (fun k => ‖⟪v k, y⟫_ℂ‖ ^ 2) (‖y

Proved

BookProof.ChapterOrthogonalSums.hasSum_smul_of_hasSum_norm_sq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterOrthogonalSums.

BookProof.ChapterOrthogonalSums.hasSum_smul_of_hasSum_norm_sq [CompleteSpace E] {ι : Type*} {v : ι → E} (hv : Orthonormal ℂ v) {y : E} (h : HasSum (fun k => ‖⟪v k, y⟫_ℂ‖ ^ 2) (‖y‖ ^ 2)) : HasSum (fun k => ⟪v k, y⟫_ℂ • v k) y

Formalization note: Lean 4 identifier BookProof.ChapterOrthogonalSums.hasSum_smul_of_hasSum_norm_sq.

Preamble

-- Generated from ChapterOrthogonalSums.lean — theorem BookProof.ChapterOrthogonalSums.hasSum_smul_of_hasSum_norm_sq
import Mathlib
import Definitions.Def_ChapterOrthogonalSums
open BookProof.ChapterOrthogonalSums


open scoped InnerProductSpace


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

Formal statement

theorem BookProof.ChapterOrthogonalSums.hasSum_smul_of_hasSum_norm_sq [CompleteSpace E] {ι : Type*} {v : ι → E}
    (hv : Orthonormal ℂ v) {y : E}
    (h : HasSum (fun k => ‖⟪v k, y⟫_ℂ‖ ^ 2) (‖y‖ ^ 2)) :
    HasSum (fun k => ⟪v k, y⟫_ℂ • v k) y := by sorry

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