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Wedge (in a vector space)

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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 $K-K=X$.


Comments

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://www.encyclopediaofmath.org/index.php?title=Wedge_(in_a_vector_space)&oldid=38845
This article was adapted from an original article by B.Z. Vulikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article