English
A non-unital subalgebra of a non-unital seminormed ring carries a restricted norm making it a non-unital seminormed ring.
Русский
Неединичное подполье над неединичнымseminormedring наследует ограниченную норму, образуя неединую нормированную полугруппу.
LaTeX
$$$\text{If } s\subseteq E\text{ is a nonunitalSubalgebra of a nonunitalSeminormedRing } E,\text{ then } s\text{ with the restricted norm is a nonunitalSeminormedRing}. $$$
Lean4
/-- A non-unital subalgebra of a non-unital seminormed ring is also a non-unital seminormed ring,
with the restriction of the norm. -/
instance nonUnitalSeminormedRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*} [NonUnitalSeminormedRing E] [Module 𝕜 E]
(s : NonUnitalSubalgebra 𝕜 E) : NonUnitalSeminormedRing s :=
{ s.toSubmodule.seminormedAddCommGroup, s.toNonUnitalRing with norm_mul_le a b := norm_mul_le a.1 b.1 }
