opNNNorm_le_iff — Mathlib

English

The NN-norm of a product equals the maximum of the NN-norms of the factors.

Русский

NN-норма произведения равно максимуму NN-норм сомножителей.

LaTeX

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

Lean4

theorem opNNNorm_le_iff {f : ContinuousMultilinearMap 𝕜 E G} {C : ℝ≥0} : ‖f‖₊ ≤ C ↔ ∀ m, ‖f m‖₊ ≤ C * ∏ i, ‖m i‖₊ := by
  simp only [← NNReal.coe_le_coe]; simp [opNorm_le_iff C.coe_nonneg, NNReal.coe_prod]

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