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# Products and Direct Sums in Locally Convex Cones

Published:2011-09-15
Printed: Dec 2012
• M. R. Motallebi,
Department of Mathematics, University of Mohaghegh Ardabili, P. O. Box 179, Ardabil, Iran
• H. Saiflu,
Department of Mathematics, Tabriz University, Tabriz, Iran
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## Abstract

In this paper we define lower, upper, and symmetric completeness and discuss closure of the sets in product and direct sums. In particular, we introduce suitable bases for these topologies, which leads us to investigate completeness of the direct sum and its components. Some results obtained about $X$-topologies and polars of the neighborhoods.
 Keywords: product and direct sum, duality, locally convex cone
 MSC Classifications: 20K25 - Direct sums, direct products, etc. 46A30 - Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness) 46A20 - Duality theory