Search

UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).

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.

In this paper we define nearly continuity and nearly openness of linear operators between locally convex cones. Also we give some conditions on locally convex cones and linear operators which ensure that every nearly continuous (nearly open) mapping is continuous (open). We show by an example that a nearly continuous operator is not necessarily continuous.

We define the concept of completion for locally convex cones. We show that how a locally convex cone with (SP) can be embedded as an upper dense subcone in an upper complete locally convex cone with (SP). We prove that every upper complete locally convex cone with (SP) is also symmetric complete.

In this paper, we define dual (abstract) neighborhood systems for locally convex cones. Also we consider three types of different polars and study some relations of them with bounded sets in locally convex cones.

In this paper we define bornological and b-bornological convergence in locally convex cones. We give some characterizations for these convergences. In special case of locally convex topological vector spaces, both these concepts reduce to the known concept of Mackey-convergence. We investigate the relationship between bornological cones and bornological convergence. Finally, we investigate bornological completeness.

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.

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 convex 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.

We define and study the projective and inductive limit notions for locally convex cones. We use convex quasiuniform structure method for this purpose. Also we study the barreledness in the locally convex cones and introduce the notion upper-barreled cones and prove that the inductive limit of upper-barreled cones is upper-barreled.

C. Park in his paper (C. Park, Isomorphisms between C ∗ -ternary algebras ,J. Math. Anal. Appl. 327 (2007) 101-115.), has proved the Hyers-Ulam-Rassias stability of homomor- phisms in C ∗ -ternary algebras and of derivations on C ∗ -ternary algebras for the following Cauchy-Jensen additive mappings: f x + y