TY - JOUR

T1 - ON EQUI-CONTINUITY OF BOUNDED ON ORDER INTERVALS FAMILIES OF SEMI-NORMS

AU - Gorokhova, Svetlana G.

AU - Emelyanov, Eduard Yu.

AU - Storozhuk, Konstantin V.

N1 - Gorokhova, S. G., Emelyanov, E. Y. and Storozhuk, K. V. On Equi-Continuity of Bounded on Order Intervals Families of Semi-Norms, Vladikavkaz Math. J., 2025, vol. 28, no. 2, pp. 35–39. DOI: 10.46698/q3483-1455-3369-u

PY - 2026

Y1 - 2026

N2 - 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.

AB - 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.

KW - equi-continuous family of semi-norms

KW - order interval

KW - ordered Banach space

KW - упорядоченное банахово пространство

KW - порядковый интервал

KW - эквинепрерывное семейство полунорм

UR - https://www.mendeley.com/catalogue/acc4c5bd-a872-381b-8f77-421b3b0f3fd8/

UR - https://www.scopus.com/pages/publications/105043744714

U2 - 10.46698/q3483-1455-3369-u

DO - 10.46698/q3483-1455-3369-u

M3 - Article

VL - 28

SP - 35

EP - 39

JO - Владикавказский математический журнал

JF - Владикавказский математический журнал

SN - 1683-3414

IS - 2

ER -