Some applications of the open mapping theorem in locally convex cones

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).

Scalar extensions for polar topologies in locally convex cones

We extend the scalar multiplications for dual pairs of cones and define the corresponding modular neighborhoods and linear polar topologies in locally convex cones. Endowed with the polar topology, every cone may be embedded in a larger cone carrying a linear polar topology over the extended scalars and the embedding is an isomorphism.

Pointwise well-posedness and scalarization of optimization problems for locally convex cone-valued functions

We investigate the pointwise well-posedness of optimization problems for locally convex conevalued functions and establish some relations between the kinds of well-posedness. Via the neighborhoods and elements, we define the scalarization functions for locally convex cones and discuss their properties. We consider the scalar optimization problems and obtain some results about the well-posedness of the optimization problems.

Weak compactness of direct sums in locally convex cones

We discuss the weakly compact subsets of direct sum cones for the upper, lower and symmetric topologies and investigate the X-topologies of the weak upper, lower and sym-metric compact subsets of direct sum cones on product cones.

Порядково борнологические локально выпуклые решеточные конусы

In this paper, we introduce the concepts of $us$-lattice cones and order bornological locally convex lattice cones. In the special case of locally convex solid Riesz spaces, these concepts reduce to the known concepts of seminormed Riesz spaces and order bornological Riesz spaces, respectively. We define solid sets in locally convex cones and present some characterizations for order bornological locally convex lattice cones.