Isac's Cones in General Vector Spaces

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Isac’S Cones In Locally Convex Spaces

First of all, we remember some usual notions and results concerning the linear (topological) spaces. A linear topology on a real or complex vector space means any topology on with respect to which the addition and the scalar multiplication are continuous, the couple being named a topological vector space. If one denotes or , then any function satisfying the following properties:

for all , (the absolute homogeneity) and

(the triangle inequality) is called seminorm on . Every linear topology is a locally convex topology iff it is generated by a family of seminorms as follows: let

be a family of seminorms defined on . For every and let

Then, the family

has the next properties:

V₀(x);V₀(x),

Key Terms in this Chapter

Topology: Every collection of sets containing the empty and the environment space, closed to the infinite unions and the finite inclusions.

Seminorm: Any positive real function defined on a real or complex vector space, satisfying the absolute homogeneity property and the triangle inequality.

Isac’s (Nuclear or Supernormal) Cone: Each convex cone in any Hausdorff locally convex space with every seminorm majorized by at least a linear and continuous functional on it.

Hausdorff Locally Convex Space: Any topological vector space endowed with the topology generated by a family of seminorms, with any two different points separated by two disjoint neighborhoods.

Minkowski Functional: Every function associated to any non-emty subset of a vector space and defined by for each ).

Choquet Boundary: The boundary of any non-empty and compact subset in every Hausdorff locally convex space, with respect to an appropriate convex cone of real functions defined on it.

Convex Cone: Every non-empty subset of a vector space closed with respect to its addition and multiplication by positive scalars operations.

Efficiency: The quality of a point in a non-void set of an ordered vector space to be of minimum (maximum) in relation with a convex cone or to its appropriate translations.

Topological Vector Space: Each vector space with its basis definition operations continuous.