norm_add_eq_sqrt_iff_real_inner_eq_zero

English

Equality of norms for sum/difference equivalent to inner product zero: ||x+y||^2 − ||x−y||^2 = 4 Re⟪x,y⟫.

Русский

Эквивалентность налаживания через квадраты норм: ||x+y||^2 − ||x−y||^2 = 4 Re⟪x,y⟫.

LaTeX

$$$\|x+y\|^2 - \|x-y\|^2 = 4\,\operatorname{Re}\langle x,y\rangle$$$

Lean4

/-- Pythagorean theorem, if-and-if vector inner product form using square roots. -/
theorem norm_add_eq_sqrt_iff_real_inner_eq_zero {x y : F} : ‖x + y‖ = √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) ↔ ⟪x, y⟫_ℝ = 0 := by
  rw [← norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero, eq_comm, sqrt_eq_iff_mul_self_eq, eq_comm] <;>
    positivity

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