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| | ''convex cone'' | | ''convex cone'' |
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| − | A convex set in a vector space over an ordered field that is invariant with respect to the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973602.png" />, that is, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973603.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973605.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973606.png" />. A convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973607.png" /> satisfying the condition: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973608.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w0973609.png" />, is called a proper (convex) cone. Each proper cone induces in the vector space a quasi-order: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736010.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736011.png" />. A proper cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736012.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736013.png" /> is called reproducing (or a domain of positivity) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736014.png" />. | + | A convex set in a vector space over an ordered field that is invariant with respect to the transformation $x\mapsto\lambda x$, $\lambda\geq0$, that is, a set $K$ such that if $x,y\in K$ and $\lambda,\mu\geq0$, then $\lambda x+\mu y\in K$. A convex cone $K$ satisfying the condition: If $x,-x\in K$ then $x=0$, is called a proper (convex) cone. Each proper cone induces in the vector space a quasi-order: $x\geq y$ when $x-y\in K$. A proper cone $K$ in a space $X$ is called reproducing (or a domain of positivity) if $K-K=X$. |
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| | ====Comments==== | | ====Comments==== |
| − | Open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736015.png" /> defined, as above, by the requirements that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736016.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736017.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736018.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736019.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736020.png" />, are usually called (open) cones. | + | Open sets $K$ defined, as above, by the requirements that $x\in K$ implies $\lambda x\in K$, for all $\lambda\geq0$, and $x,y\in K$ implies $x+y\in K$, are usually called (open) cones. |
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| − | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736021.png" /> be an open cone in a real topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736022.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736023.png" /> be the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736024.png" /> with a bounded open ball with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736025.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736026.png" /> be an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736027.png" />. Let | + | Let $K$ be an open cone in a real topological vector space $k$, let $V$ be the intersection of $K$ with a bounded open ball with centre at $0\in k$, and let $U\neq\emptyset$ be an open set in $k$. Let |
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| − | <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/w/w097/w097360/w09736028.png" /></td> </tr></table>
| + | $$W^+=U+iV,\quad W^-=U-iV$$ |
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| − | be sets in the complexification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736030.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736032.png" /> are called wedges (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097360/w09736033.png" /> is their "edge" ). For a use of wedges in the theory of analytic continuation see [[Bogolyubov theorem|Bogolyubov theorem]]. | + | be sets in the complexification of $k$ ($i^2=-1$). Then $W^+$ and $W^-$ are called wedges (and $U+i0$ is their "edge"). For a use of wedges in the theory of analytic continuation see [[Bogolyubov theorem|Bogolyubov theorem]]. |
Revision as of 13:31, 31 July 2014
convex cone
A convex set in a vector space over an ordered field that is invariant with respect to the transformation $x\mapsto\lambda x$, $\lambda\geq0$, that is, a set $K$ such that if $x,y\in K$ and $\lambda,\mu\geq0$, then $\lambda x+\mu y\in K$. A convex cone $K$ satisfying the condition: If $x,-x\in K$ then $x=0$, is called a proper (convex) cone. Each proper cone induces in the vector space a quasi-order: $x\geq y$ when $x-y\in K$. A proper cone $K$ in a space $X$ is called reproducing (or a domain of positivity) if $K-K=X$.
Open sets $K$ defined, as above, by the requirements that $x\in K$ implies $\lambda x\in K$, for all $\lambda\geq0$, and $x,y\in K$ implies $x+y\in K$, are usually called (open) cones.
Let $K$ be an open cone in a real topological vector space $k$, let $V$ be the intersection of $K$ with a bounded open ball with centre at $0\in k$, and let $U\neq\emptyset$ be an open set in $k$. Let
$$W^+=U+iV,\quad W^-=U-iV$$
be sets in the complexification of $k$ ($i^2=-1$). Then $W^+$ and $W^-$ are called wedges (and $U+i0$ is their "edge"). For a use of wedges in the theory of analytic continuation see Bogolyubov theorem.
How to Cite This Entry:
Wedge (in a vector space). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wedge_(in_a_vector_space)&oldid=12115
This article was adapted from an original article by B.Z. Vulikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article