with_bound — Mathlib

English

The norm of the restricted-bilinear map equals the original norm, reinforcing identity of shortening/scalar restriction.

Русский

Нормa ограниченного билинейного отображения равна исходной норме.

LaTeX

$$$ \\|B.bilinearRestrictScalars 𝕜\\| = \\|B\\| $$$

Lean4

theorem with_bound {f : E → F} (hf : IsLinearMap 𝕜 f) (M : ℝ) (h : ∀ x : E, ‖f x‖ ≤ M * ‖x‖) : IsBoundedLinearMap 𝕜 f :=
  ⟨hf,
    by_cases
      (fun (this : M ≤ 0) =>
        ⟨1, zero_lt_one, fun x => (h x).trans <| mul_le_mul_of_nonneg_right (this.trans zero_le_one) (norm_nonneg x)⟩)
      fun (this : ¬M ≤ 0) => ⟨M, lt_of_not_ge this, h⟩⟩

Похожие утверждения