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Lines and Hyperplanes associated with Families of Closed and Bounded Sets in Conjugate Banach Spaces

Published online by Cambridge University Press:  20 November 2018

M. Edelstein
M. Edelstein*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 933 - 938
Copyright
Copyright © Canadian Mathematical Society 1970

References

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