In this paper, we verify a relation between the symmetric continuity of lower semicontinuous extended seminorms on a locally convex cone with respect to the symmetric topology and barreledness of the locally convex cone. The assertion, whether a barreled cone is upper-barreled or not, was posed as a question in Ranjbari and Saiflu (J Math Anal Appl 332(2):1097–1108, 2007). We show that a barreled locally convex cone is not necessarily upper-barreled.

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关于局部凸锥上的桶状度的一些注意事项

在本文中，我们验证了局部凸锥上的下半连续扩展半范数的对称连续性与局部凸锥的对称拓扑和桶形度之间的关系。兰吉巴里（Ranjbari）和赛弗（Saiflu）提出了一个问题，即桶形圆锥是否是上桶形的（J Math Anal Appl 332（2）：1097-1108，2007）。我们显示桶形的局部凸锥不一定是上桶形的。