uniformEquicontinuous_restrict_iff

English

The restrict-then-equicontinuity equivalence under a finite index holds for uniform equicontinuity as well.

Русский

Униформная эконтинуальность сохраняется при ограничении конечного индекса.

LaTeX

$$$$ \operatorname{uniformEquicontinuous}(F) \iff \operatorname{uniformEquicontinuousOn}(F, S). $$$$

Lean4

theorem uniformEquicontinuous_restrict_iff (F : ι → β → α) {S : Set β} :
    UniformEquicontinuous (S.restrict ∘ F) ↔ UniformEquicontinuousOn F S :=
  by
  rw [UniformEquicontinuous, UniformEquicontinuousOn]
  conv in _ ⊓ _ => rw [← Subtype.range_val (s := S), ← range_prodMap, ← map_comap]
  rfl

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