`BookProof.ChapterH7.inner_self_real` (X : E →L[ℂ] E) (hX : IsSelfAdjoint X) (x : E) : (inner ℂ x (X x) : ℂ).im = 0

Proved

BookProof.ChapterH7.inner_self_real

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterH7.

BookProof.ChapterH7.inner_self_real (X : E →L[ℂ] E) (hX : IsSelfAdjoint X) (x : E) : (inner ℂ x (X x) : ℂ).im = 0

Formalization note: Lean 4 identifier BookProof.ChapterH7.inner_self_real.

Preamble

-- Generated from ChapterH7.lean — theorem BookProof.ChapterH7.inner_self_real
import Definitions.Def_ChapterH4
import Definitions.Def_ChapterH6
import Mathlib
import Definitions.Def_ChapterH7
open BookProof.ChapterH7


noncomputable section

open BookProof.ChapterH4 BookProof.ChapterH6

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

Formal statement

theorem BookProof.ChapterH7.inner_self_real (X : E →L[ℂ] E) (hX : IsSelfAdjoint X) (x : E) :
    (inner ℂ x (X x) : ℂ).im = 0 := by sorry

View graph