Positive cone

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

1) if $ x, y \in K $ and $ \alpha , \beta \geq 0 $, then $ \alpha x + \beta y \in K $;

2) $ K \cap (- K) = \{ 0 \} $.

A positive cone defines a pre-order in $ E $ by putting $ x \prec y $ if $ y - x \in K $. (This pre-order is compatible with the vector space operations.)

Let $ E $ be a Banach space. The cone $ K $ is a closed reproducing positive cone if for all $ z \in E $ there are $ x, y \in K $ such that $ z = x- y $. In that case there is a constant $ M $ independent of $ z $ such that there always exist $ x, y $ such that $ z = x- y $ with $ \| x \| + \| y \| \leq M \| z \| $. A solid positive cone, i.e. one having interior points, is reproducing.

Let $ E ^ {*} $ be the dual of the Banach space $ E $. If $ K \subset E $ is a closed reproducing positive cone, then the set $ K ^ {*} \subset E ^ {*} $ of positive functionals (with respect to the positive cone, i.e. those $ f $ such that $ f( x) \geq 0 $ for $ x \in K $) is also a positive cone (this is the so-called conjugate cone). The positive cone $ K $ can be recovered from $ K ^ {*} $, namely:

$$ K = \{ {x \in E } : {f( x) \geq 0 \textrm{ for } f \in K ^ {*} } \} . $$

If $ K $ is a solid positive cone, then its interior coincides with

$$ \{ {x \in E } : {f( x) > 0 \textrm{ for } f \in K ^ {*} , f \neq 0 } \} . $$

A cone in the Banach space $ E $ is called normal if one can find a $ \delta > 0 $ so that $ \| x + y \| \geq \delta ( \| x \| + \| y \| ) $ for $ x, y \in K $. A positive cone is normal if and only if the conjugate cone $ K ^ {*} $ is reproducing. If $ K $ is a reproducing cone, then the conjugate cone $ K ^ {*} $ is normal.

A cone $ K $ is called a lattice cone if each pair of elements $ x, y \in E $ has a least upper bound $ z = \sup ( x, y) $, i.e. $ z \geq x, y $ and for any $ z _ {1} \in E $ it follows from $ z _ {1} \geq x, y $ that $ z _ {1} \geq z $. If a positive cone is regular and lattice, then any countable bounded subset has a least upper bound.

References

[1]	M.A. Krasnosel'skii, "Positive solutions of operator equations" , Wolters-Noordhoff (1964) (Translated from Russian)

References

[a1]	H.H. Schaefer, "Banach lattices and positive operators" , Springer (1974)
[a2]	A.C. Zaanen, W. Luxemburg, "Riesz spaces" , I , North-Holland (1983)