Difference between revisions of "Positive cone"
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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. | ||
| − | If | + | 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 | ||
| − | A cone | + | $$ |
| + | \{ {x \in E } : {f( x) > 0 \textrm{ for } f \in K ^ {*} , f \neq 0 } \} | ||
| + | . | ||
| + | $$ | ||
| + | |||
| + | 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=15148
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