uniformEquicontinuousOn_finite — Mathlib

English

If ι is finite, uniform equicontinuity on S is equivalent to uniform continuity of each F_i on S.

Русский

Если ι конечно, униформная эконтинуальность на S эквивалентна униформной непрерывности каждого F_i на S.

LaTeX

$$$$ [Finite ι] \ operatorname{UniformEquicontinuousOn} F S \iff \forall i, UniformContinuousOn (F i) S. $$$$

Lean4

theorem uniformEquicontinuousOn_finite [Finite ι] {F : ι → β → α} {S : Set β} :
    UniformEquicontinuousOn F S ↔ ∀ i, UniformContinuousOn (F i) S := by
  simp only [UniformEquicontinuousOn, eventually_all, @forall_swap _ ι]; rfl

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