Abstract: We investigate conditions which provide equi-continuity of bounded on order intervals families of semi-norms. This investigation develops further recent results on uniform boundedness of bounded on order intervals families of linear operators from an ordered Banach space with a closed generating cone to a normed space. More precisely, we prove that a family of semi-norms on an ordered Banach space with a closed generating cone is equi-continuous if it is equi-bounded on order intervals. In particular, every semi-norm on an ordered Banach space with a closed generating cone is continuous, whenever it is bounded on order intervals. It leads to straightforward proofs of above-mentioned recent results on automatic continuity of certain linear operators and on uniform boundedness of some families of linear operators. Furthermore, we prove that every order-to-topology bounded set of linear operators from an ordered Banach space with a closed generating cone to a locally convex space is equi-continuous.

Cite: Gorokhova S.G. , Emelyanov E.Y. , Storozhuk K.V.
On Equi-Continuity of Bounded on Order Intervals Families of Semi-Norms
Владикавказский математический журнал (Vladikavkaz Mathematical Journal). 2026. V.28. N2(110). P.35-39. DOI: 10.46698/q3483-1455-3369-u Scopus OpenAlex

Dates:

Submitted:

Nov 19, 2025

Identifiers:

≡ Scopus:	2-s2.0-105043744714
≡ OpenAlex:	W7165538036

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