English
If the norms ‖f(i)‖₊ are pairwise distinct on a finite set, the NNNorm of the product equals the supremum of the NNNorms.
Русский
Если нормы ‖f(i)‖₊ различны на конечном множестве, NNNorm произведения равно supnormNNNorm норм элементов.
LaTeX
$$$\\text{If } s \\neq \\emptyset \\text{ and } (s.toSet).Pairwise (\\|f(i)\\|_{+}) \\Rightarrow \\|\\prod_{i\\in s} f(i)\\|_{nn} = s.sup' (\\|f(i)\\|_{nn})$$$
Lean4
/-- A homomorphism `f` of seminormed groups is Lipschitz, if there exists a constant `C` such that
for all `x`, one has `‖f x‖ ≤ C * ‖x‖`. The analogous condition for a linear map of
(semi)normed spaces is in `Mathlib/Analysis/NormedSpace/OperatorNorm.lean`. -/
@[to_additive /-- A homomorphism `f` of seminormed groups is Lipschitz, if there exists a constant
`C` such that for all `x`, one has `‖f x‖ ≤ C * ‖x‖`. The analogous condition for a linear map of
(semi)normed spaces is in `Mathlib/Analysis/NormedSpace/OperatorNorm.lean`. -/
]
theorem lipschitz_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
LipschitzWith (Real.toNNReal C) f :=
LipschitzWith.of_dist_le' fun x y => by simpa only [dist_eq_norm_div, map_div] using h (x / y)
