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|Title:||The Semi-Kadec-Klee Condition and Nearest-Point Properties of Sets in Normed Linear Spaces|
|Author(s):||Megginson, Robert Eugene|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||The first part of this thesis studies the relationship between convexity and nearest-point properties of sets in certain normed linear spaces. Here are some typical results.
A normed linear space X is very rotund if every support functional in the dual unit sphere is a point of smoothness of the dual unit ball. It is shown that a normed space is very rotund if and only if each of its proximinal convex sets is an approximately weakly compact Chebyshev set.
A normed linear space X is a semi-Kadec-Klee space if, whenever a sequence (x(,n)) and a point x on the unit sphere are such that (x(,n)) converges weakly to x, and (H(,n)) is a sequence of hyperplanes such that H(,n) supports the unit ball at x(,n), then the distance from H(,n) to x tends to zero. Many results in the literature state that under certain rather strong geometric conditions on a space, the nonempty closed convex sets are exactly the Chebyshev sets having some particular property. We show that many of these results remain true if the space is a rotund, reflexive, smooth semi-Kadec-Klee space, and that this geometric hypothesis cannot be made much weaker.
The second part of the thesis studies the semi-Kadec-Klee property in more detail. It is shown that this property is equivalent to a continuity property of the norm-duality map J. The connection between the semi-Kadec-Klee property and supportive compactness is examined, where a set M in a space X is defined to be supportively compact if certain images J(,x)(M) of M in X* are approximately compact with respect to the origin.
Two appendices contain material tangentially related to the main theme of the thesis. The first appendix shows that many results from this branch of approximation theory that have been treated as corollaries of James's theorem can actually be obtained from more basic principles. In addition, an elementary proof of James's theorem applicable to a large class of spaces is given. The second appendix contains material about the approximative compactness of closed balls.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984.
|Date Available in IDEALS:||2014-12-16|