`BookProof.ChapterNumericalRangeCrouzeix.inner_sub_const_smul` (A : E →L[ℂ] E) (c : ℂ) (x : E) : (⟪ x, (A - c • (1 : E →L[ℂ] E)) x ⟫_ℂ) = ⟪ x, A x ⟫_ℂ - c * (‖x‖ ^ 2 : ℝ)

Proved

BookProof.ChapterNumericalRangeCrouzeix.inner_sub_const_smul

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeCrouzeix.

BookProof.ChapterNumericalRangeCrouzeix.inner_sub_const_smul (A : E →L[ℂ] E) (c : ℂ) (x : E) : (⟪ x, (A - c • (1 : E →L[ℂ] E)) x ⟫_ℂ) = ⟪ x, A x ⟫_ℂ - c * (‖x‖ ^ 2 : ℝ)

Formalization note: Lean 4 identifier BookProof.ChapterNumericalRangeCrouzeix.inner_sub_const_smul.

Preamble

-- Generated from ChapterNumericalRangeCrouzeix.lean — theorem BookProof.ChapterNumericalRangeCrouzeix.inner_sub_const_smul
import Mathlib
import Definitions.Def_ChapterNumericalRangeCrouzeix
open BookProof.ChapterNumericalRangeCrouzeix


open scoped InnerProductSpace


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

Formal statement

theorem BookProof.ChapterNumericalRangeCrouzeix.inner_sub_const_smul (A : E →L[ℂ] E) (c : ℂ) (x : E) :
    (⟪ x, (A - c • (1 : E →L[ℂ] E)) x ⟫_ℂ) = ⟪ x, A x ⟫_ℂ - c * (‖x‖ ^ 2 : ℝ) := by sorry

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