Locally Convex Cones (A Generalization of Locally Convex Topological Vector Spaces)

Locally convex  cones were proposed for applications in approximation theory but have matured into a mathematical theory. The aim of this book is to offer a coherent presentation of its basic principles and later developments and to establish a consistent reference source.

The build of the theory is modelled after the cone of convex subsets of a locally convex vector space. It does not satisfy the cancellation law and cannot be embedded into a vector space. The topology of the vector space leads to the definition of three cone topologies. Similar settings occur in integration theory, potential theory, and other mathematical areas. Locally convex topological vector spaces are special cases of locally convex cones.

Some of the notions for locally convex cones remain close to those for topological vector spaces. The dual cone consists of all continuous linear functionals that can take infinite values. This yields a rich duality theory, including Hahn-Banach type extension and separation theorems.  Other concepts, such as boundedness and connectedness, extend into different territories. This book serves as a primer for students and a reference tool for researchers.