unit_le_opNorm — Mathlib

English

The NN-norm of the product of two multilinear maps equals the maximum of their NN-norms.

Русский

NN-норма произведения двух мультилейных отображений равна максимуму их NN-норм.

LaTeX

$$$$ \\|f.prod g\\|_{nn} = \\max\\{ \\|f\\|_{nn}, \\|g\\|_{nn} \\}.$$$$

Lean4

/-- The image of the unit ball under a continuous multilinear map is bounded. -/
theorem unit_le_opNorm (f : ContinuousMultilinearMap 𝕜 E G) {m : ∀ i, E i} (h : ‖m‖ ≤ 1) : ‖f m‖ ≤ ‖f‖ :=
  (le_opNorm_mul_pow_card_of_le f h).trans <| by simp

Похожие утверждения