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A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738601.png" /> of a real [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738602.png" /> satisfying the following conditions:
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1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738604.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738605.png" />;
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{{TEX|auto}}
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{{TEX|done}}
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738606.png" />.
+
A subset  $  K $
 +
of a real [[Vector space|vector space]]  $  E $
 +
satisfying the following conditions:
  
A positive cone defines a pre-order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738607.png" /> by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738608.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p0738609.png" />. (This pre-order is compatible with the vector space operations.)
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1) if  $  x, y \in K $
 +
and  $  \alpha , \beta \geq  0 $,
 +
then  $  \alpha x + \beta y \in K $;
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386010.png" /> be a [[Banach space|Banach space]]. The cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386011.png" /> is a closed reproducing positive cone if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386012.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386014.png" />. In that case there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386015.png" /> independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386016.png" /> such that there always exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386018.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386019.png" />. A solid positive cone, i.e. one having interior points, is reproducing.
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2)  $  K \cap (- K) = \{ 0 \} $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386020.png" /> be the dual of the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386022.png" /> is a closed reproducing positive cone, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386023.png" /> of positive functionals (with respect to the positive cone, i.e. those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386026.png" />) is also a positive cone (this is the so-called conjugate cone). The positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386027.png" /> can be recovered from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386028.png" />, namely:
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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.)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386029.png" /></td> </tr></table>
+
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.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386030.png" /> is a solid positive cone, then its interior coincides with
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386031.png" /></td> </tr></table>
+
$$
 +
= \{ {x \in E } : {f( x) \geq  0 \textrm{ for }  f \in K  ^ {*} } \}
 +
.
 +
$$
  
A cone in the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386032.png" /> is called normal if one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386033.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386034.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386035.png" />. A positive cone is normal if and only if the conjugate cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386036.png" /> is reproducing. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386037.png" /> is a reproducing cone, then the conjugate cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386038.png" /> is normal.
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If  $  K $
 +
is a solid positive cone, then its interior coincides with
  
A cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386039.png" /> is called a lattice cone if each pair of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386040.png" /> has a least upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386041.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386042.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386043.png" /> it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386044.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073860/p07386045.png" />. If a positive cone is regular and lattice, then any countable bounded subset has a least upper bound.
+
$$
 +
\{ {x \in E } : {f( x) > 0  \textrm{ for }  f \in K  ^ {*} , f \neq 0 } \}
 +
.
 +
$$
 +
 
 +
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:07, 6 June 2020


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)

Comments

References

[a1] H.H. Schaefer, "Banach lattices and positive operators" , Springer (1974)
[a2] A.C. Zaanen, W. Luxemburg, "Riesz spaces" , I , North-Holland (1983)
How to Cite This Entry:
Positive cone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_cone&oldid=48252
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article