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Wedge (in a vector space)

From Encyclopedia of Mathematics
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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 , , that is, a set such that if and , then . A convex cone satisfying the condition: If then , is called a proper (convex) cone. Each proper cone induces in the vector space a quasi-order: when . A proper cone in a space is called reproducing (or a domain of positivity) if .


Comments

Open sets defined, as above, by the requirements that implies , for all , and implies , are usually called (open) cones.

Let be an open cone in a real topological vector space , let be the intersection of with a bounded open ball with centre at , and let be an open set in . Let

be sets in the complexification of (). Then and are called wedges (and 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=32612
This article was adapted from an original article by B.Z. Vulikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article