uniformSpace_eq — Mathlib

English

The uniform space on UniformConvergenceCLM σ F 𝔖 equals the comap of the standard uniform space along the evaluation map.

Русский

Унитепная пространство на UniformConvergenceCLM σ F 𝔖 равно выписке по отображению оценки.

LaTeX

$$$\\operatorname{instUniformSpace}(\\sigma,F,\\mathcal{S}) = \\mathrm{UniformSpace.comap}(\\mathrm{UniformOnFun.ofFun}\\,\\mathcal{S} \\circ \\mathrm{DFunLike.coe}, \\ \\mathrm{UniformOnFun.uniformSpace} E F 𝔖).$$$

Lean4

theorem uniformSpace_eq [UniformSpace F] [IsUniformAddGroup F] (𝔖 : Set (Set E)) :
    instUniformSpace σ F 𝔖 =
      UniformSpace.comap (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe) (UniformOnFun.uniformSpace E F 𝔖) :=
  by rw [instUniformSpace, UniformSpace.replaceTopology_eq]

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