uniformEquicontinuous_of_equicontinuous

English

An equicontinuous family of functions defined on a compact space is automatically uniformly equicontinuous.

Русский

Еквиподдисциплины экоинконтинуальные семейства функций на компактном пространстве автоматически равномерно экоинконтинуальны.

LaTeX

$$\\text{Equicontinuous}(F) \\Rightarrow \\text{UniformEquicontinuous}(F) \\text{ when } \\text{CompactSpace}(β)$$

Lean4

/-- An equicontinuous family of functions defined on a compact uniform space is automatically
uniformly equicontinuous. -/
theorem uniformEquicontinuous_of_equicontinuous {ι : Type*} {F : ι → β → α} [CompactSpace β] (h : Equicontinuous F) :
    UniformEquicontinuous F := by
  rw [equicontinuous_iff_continuous] at h
  rw [uniformEquicontinuous_iff_uniformContinuous]
  exact CompactSpace.uniformContinuous_of_continuous h

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