# Positive cone

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A subset of a real vector space satisfying the following conditions:

1) if and , then ;

2) .

A positive cone defines a pre-order in by putting if . (This pre-order is compatible with the vector space operations.)

Let be a Banach space. The cone is a closed reproducing positive cone if for all there are such that . In that case there is a constant independent of such that there always exist such that with . A solid positive cone, i.e. one having interior points, is reproducing.

Let be the dual of the Banach space . If is a closed reproducing positive cone, then the set of positive functionals (with respect to the positive cone, i.e. those such that for ) is also a positive cone (this is the so-called conjugate cone). The positive cone can be recovered from , namely: If is a solid positive cone, then its interior coincides with A cone in the Banach space is called normal if one can find a so that for . A positive cone is normal if and only if the conjugate cone is reproducing. If is a reproducing cone, then the conjugate cone is normal.

A cone is called a lattice cone if each pair of elements has a least upper bound , i.e. and for any it follows from that . If a positive cone is regular and lattice, then any countable bounded subset has a least upper bound.

How to Cite This Entry:
Positive cone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_cone&oldid=15148
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article