homothety_inverse — Mathlib

English

The continuous linear map produced from exists-bound construction agrees with f when viewed as a linear map.

Русский

Полученное непрерывное линейное отображение совпадает с исходной линейной картой.

LaTeX

$$$(f.mkContinuousOfExistsBound h : E \to_{σ}^L F) = f$$$

Lean4

theorem homothety_inverse (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) :
    (∀ x : E, ‖f x‖ = a * ‖x‖) → ∀ y : F, ‖f.symm y‖ = a⁻¹ * ‖y‖ :=
  by
  intro hf y
  calc
    ‖f.symm y‖ = a⁻¹ * (a * ‖f.symm y‖) := by rw [← mul_assoc, inv_mul_cancel₀ (ne_of_lt ha).symm, one_mul]
    _ = a⁻¹ * ‖f (f.symm y)‖ := by rw [hf]
    _ = a⁻¹ * ‖y‖ := by simp

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