(T : F →L[ℂ] F) (c : ℝ) (x : F) : (inner ℂ ((c : ℂ) • x) (T ((c : ℂ) • x)) : ℂ).re = c ^ 2 * (inner ℂ x (T x) : ℂ).re

-- Generated from ChapterSirkRitzSpectrum.lean — theorem BookProof.ChapterSirkRitzSpectrum.re_inner_real_smul
import Mathlib
import Definitions.Def_ChapterSirkRitzSpectrum
open BookProof.ChapterSirkRitzSpectrum







noncomputable section


open Filter Topology RCLike ContinuousLinearMap ComplexOrder Pointwise

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

theorem BookProof.ChapterSirkRitzSpectrum.re_inner_real_smul (T : F →L[ℂ] F) (c : ℝ) (x : F) :
    (inner ℂ ((c : ℂ) • x) (T ((c : ℂ) • x)) : ℂ).re = c ^ 2 * (inner ℂ x (T x) : ℂ).re := by sorry