Positivity (2011) 15:465–471
Strict inductive limits in locally convex cones
Received: 21 August 2010 / Accepted: 6 October 2010 / Published online: 21 October 2010
© Springer Basel AG 2010
Abstract We propose a definition for strict inductive limits in locally convex cones.
By this definition, we prove that the strict inductive limit of a sequence of locally con-
vex cones with the strict separation property has the same strict separation property.
Also we establish that the strict inductive limit of a sequences of separated cones is
separated too. Finally we verify barreledness for this strict inductive limit.
Keywords Locally convex cone · Convex quasiuniform structure · Strict inductive
Mathematics Subject Classiﬁcation (2000) 46A03 · 46A08 · 46A13
The general theory of locally convex cones as developed in  deals with preordered
cones. We review some of the main concepts and refer to or for details; for
recent researches see [3,4,6,8,9].
A cone is a set P endowed with an addition and a scalar multiplication for non-
negative real numbers. The addition is associative and commutative, and there is a
neutral element 0 ∈ P. For the scalar multiplication the usual associative and distrib-
utive properties hold. We have 1a = a and 0a = 0 for all a ∈ P. A preordered cone
(ordered cone) is a cone with a preorder, that is a reﬂexive transitive relation ≤ which
is compatible with the algebraic operations.
This research was partially supported by the University of Tabriz.
A. Ranjbari (
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran