A note on barreledness in locally convex cones

TL;DR

This paper investigates the properties of barreledness in locally convex cones, demonstrating that being barreled does not imply being upper-barreled, thus clarifying a previously posed question in the theory.

Contribution

It provides a counterexample showing that a barreled locally convex cone need not be upper-barreled, addressing an open question in the field.

Findings

Barreled cones are not necessarily upper-barreled.

Counterexample disproves the implication.

Clarifies the relationship between barreledness and upper-barreledness.

Abstract

Locally convex cones are generalization of locally convex spaces. The assertion, whether a barreled cone is an upper-barreled cone or not, was posed as a question in [A. Ranjbari, H. Saiflu, Projective and inductive limits in locally convex cones, J. Math. Anal. Appl. 332 (2) (2007) 1097-1108]. In this paper, we show that a barreled locally convex cone is not necessarily upper-barreled.