norm_sq — Mathlib

English

In a complex inner product space, ||x||^2 = Re ⟨x,x⟩.

Русский

В комплексном пространства с скалярным произведением ||x||^2 = Re ⟨x,x⟩.

LaTeX

$$$\|x\|^{2} = \operatorname{Re}\langle x,x\rangle$$$

Lean4

theorem norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) :=
  by
  simp only [inner_, normSq_apply, ofNat_re, ofNat_im, map_sub, map_add, ofReal_re, ofReal_im, mul_re, inv_re, mul_im,
    I_re, inv_im]
  have h₁ : ‖x - x‖ = 0 := by simp
  have h₂ : ‖x + x‖ = 2 • ‖x‖ := by convert norm_nsmul 𝕜 2 x using 2; module
  rw [h₁, h₂]
  ring

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