`BookProof.ChapterA.inner_self_complex_iff_real` (T : H₁ → H₂) : (∀ x, (inner ℂ (T x) (T x) : ℂ) = (inner ℂ x x : ℂ)) ↔ (∀ x, (inner ℝ (T x) (T x) : ℝ) = (inner ℝ x x : ℝ))

Proved

BookProof.ChapterA.inner_self_complex_iff_real

by leonardopedro · Oct 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1Prop5.

BookProof.ChapterA.inner_self_complex_iff_real (T : H₁ → H₂) : (∀ x, (inner ℂ (T x) (T x) : ℂ) = (inner ℂ x x : ℂ)) ↔ (∀ x, (inner ℝ (T x) (T x) : ℝ) = (inner ℝ x x : ℝ))

Formalization note: Lean 4 identifier BookProof.ChapterA.inner_self_complex_iff_real.

Preamble

-- Generated from ChapterA1Prop5.lean — theorem BookProof.ChapterA.inner_self_complex_iff_real
import Mathlib
import Definitions.Def_ChapterA1Prop5
import Definitions.Def_ChapterA
open BookProof.ChapterA

variable {H₁ H₂ : Type*} [NormedAddCommGroup H₁] [InnerProductSpace ℂ H₁]
  [NormedAddCommGroup H₂] [InnerProductSpace ℂ H₂]


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace





attribute [local instance] InnerProductSpace.rclikeToReal

Formal statement

theorem BookProof.ChapterA.inner_self_complex_iff_real (T : H₁ → H₂) :
    (∀ x, (inner ℂ (T x) (T x) : ℂ) = (inner ℂ x x : ℂ)) ↔
    (∀ x, (inner ℝ (T x) (T x) : ℝ) = (inner ℝ x x : ℝ)) := by sorry

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