Isac's Cones in General Vector Spaces

Isac's Cones in General Vector Spaces

Vasile Postolică (Romanian Academy of Scientists,“Vasile Alecsandri” University of Bacău, Faculty of Sciences, Bacău, România)
Copyright: © 2014 |Pages: 20
DOI: 10.4018/978-1-4666-5202-6.ch121

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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 978-1-4666-5202-6.ch121.m01 means any topology 978-1-4666-5202-6.ch121.m02 on 978-1-4666-5202-6.ch121.m03 with respect to which the addition and the scalar multiplication are continuous, the couple 978-1-4666-5202-6.ch121.m04 being named a topological vector space. If one denotes 978-1-4666-5202-6.ch121.m05 or 978-1-4666-5202-6.ch121.m06, then any function 978-1-4666-5202-6.ch121.m07 satisfying the following properties:

978-1-4666-5202-6.ch121.m08
for all 978-1-4666-5202-6.ch121.m09, 978-1-4666-5202-6.ch121.m10 (the absolute homogeneity) and
978-1-4666-5202-6.ch121.m11
(the triangle inequality) is called seminorm on 978-1-4666-5202-6.ch121.m12. Every linear topology is a locally convex topology iff it is generated by a family of seminorms as follows: let
978-1-4666-5202-6.ch121.m13
be a family of seminorms defined on 978-1-4666-5202-6.ch121.m14. For every 978-1-4666-5202-6.ch121.m15 and 978-1-4666-5202-6.ch121.m16let

Then, the family

978-1-4666-5202-6.ch121.m18
has the next properties:

978-1-4666-5202-6.ch121.m19V0(x);978-1-4666-5202-6.ch121.m20V0(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.

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