continuous_of_lowerSemicontinuous

English

If 0 ∈ s, balancedCore 𝕜 s equals the intersection of all r•s with 1≤‖r‖.

Русский

Если 0 ∈ s, то balancedCore 𝕜 s равняется пересечению всех r•s при 1≤‖r‖.

LaTeX

$$If\\ 0 \\in s\\ then\\ balancedCore\\ 𝕜\\ s = \\bigcap_{r:𝕜,\\ 1\\le \\|r\\|} (r\\cdot s)$$

Lean4

theorem continuous_of_lowerSemicontinuous {𝕜 E : Type*} [AddGroup E] [SMul 𝕜 E] [SeminormedRing 𝕜] [TopologicalSpace E]
    [BarrelledSpace 𝕜 E] (p : Seminorm 𝕜 E) (hp : LowerSemicontinuous p) : Continuous p :=
  BarrelledSpace.continuous_of_lowerSemicontinuous p hp

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