# 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)