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A subset  $  K $
of a real [[Vector space|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Krasnosel'skii,   "Positive solutions of operator equations" , Wolters-Noordhoff  (1964)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.H. Schaefer,   "Banach lattices and positive operators" , Springer  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,   W. Luxemburg,   "Riesz spaces" , '''I''' , North-Holland  (1983)</TD></TR></table>