English
Let {p_i} be a family of seminorms on a vector space E such that their values are bounded above. Then for every x in E, the value of the supremum seminorm at x equals the supremum of the values p_i(x): (sup_i p_i)(x) = sup_i p_i(x).
Русский
Пусть {p_i} — семейство полиномонов на векторном пространстве E such that значения ограничены сверху. Тогда для каждого x ∈ E значение супремума полиномона в точке x равно верхней граности значений p_i(x): (sup_i p_i)(x) = sup_i p_i(x).
LaTeX
$$$\\left(\\sup_i p_i\\right)(x) = \\sup_i p_i(x)$$$
Lean4
protected theorem coe_iSup_eq {ι : Sort*} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) :
↑(⨆ i, p i) = ⨆ i, ((p i : Seminorm 𝕜 E) : E → ℝ) :=
by
rw [← sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm 𝕜 E => (p : E → ℝ)) p
