cont_withSeminorms_normedSpace — Mathlib

English

The uniform structure induced by a family of seminorms p equals the infimum of the uniform structures induced by each p_i.

Русский

Унитарная структура, индуцированная семейством семинорм p, равна инфиму структур, индуцированных каждым p_i.

LaTeX

$$$WithSeminorms\\ p \\iff \\; (u = iInf (p_i).toSeminormedAddCommGroup.toUniformSpace)$$$

Lean4

theorem cont_withSeminorms_normedSpace (F) [SeminormedAddCommGroup F] [NormedSpace 𝕝₂ F] [TopologicalSpace E]
    {p : ι → Seminorm 𝕝 E} (hp : WithSeminorms p) (f : E →ₛₗ[τ₁₂] F)
    (hf : ∃ (s : Finset ι) (C : ℝ≥0), (normSeminorm 𝕝₂ F).comp f ≤ C • s.sup p) : Continuous f :=
  by
  rw [← Seminorm.isBounded_const (Fin 1)] at hf
  exact continuous_from_bounded hp (norm_withSeminorms 𝕝₂ F) f hf

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