English
The norm is bounded by sqrt(2) times the maximum of the absolute values of the real and imaginary parts: ||z|| ≤ √2 · max(|Re z|, |Im z|).
Русский
Норма modуль ограничена sqrt(2) умноженным на максимум из |Re z|, |Im z|: ||z|| ≤ √2 · max(|Re z|, |Im z|).
LaTeX
$$$$\\forall z \\in \\mathbb{C}, \\|z\\| \\le \\sqrt{2} \\cdot \\max (|\\Re z|, |\\Im z|)$$$$
Lean4
theorem norm_le_sqrt_two_mul_max (z : ℂ) : ‖z‖ ≤ √2 * max |z.re| |z.im| :=
by
obtain ⟨x, y⟩ := z
simp only [norm_def, normSq_mk, norm_def, ← sq]
set m := max |x| |y|
have hm₀ : 0 ≤ m := by positivity
calc
√(x ^ 2 + y ^ 2) ≤ √(m ^ 2 + m ^ 2) :=
by
gcongr √(?_ + ?_) <;> rw [sq_le_sq, abs_of_nonneg hm₀]
exacts [le_max_left _ _, le_max_right _ _]
_ = √2 * m := by rw [← two_mul, Real.sqrt_mul, Real.sqrt_sq] <;> positivity
