English
UniformEquicontinuousOn F S is equivalent to UniformEquicontinuousOn ((↑) : range F → β → α) S.
Русский
Униформная эконтинуированностьOn через диапазон.
LaTeX
$$$\\operatorname{UniformEquicontinuousOn} F S \\iff \\operatorname{UniformEquicontinuousOn} \\bigl((\\uparrow) : \\operatorname{range} F \\to X \\to \\alpha\\bigr) S.$$$
Lean4
/-- A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.
This is very useful for developing the equicontinuity API, but it should not be used directly
for other purposes. -/
theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
UniformEquicontinuous F ↔ UniformContinuous (ofFun ∘ Function.swap F : β → ι →ᵤ α) :=
by
rw [UniformContinuous, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff]
rfl
