This paper is a study of the relation between smoothness of the norm on a normed linear space and the property that every Chebyshev subset is convex. Every normed linear space of finite dimension, having a smooth norm, has the property that every Chebyshev subset is convex. In the second chapter two properties of the norm, uniform Gateaux differentiability and uniform Frechet differentiability where the latter implies the former, are given and are shown to be equivalent to smoothness of the norm in spaces of finite dimension. In the third chapter it is shown that every reflexive normed linear space having a uniformly Gateaux differentiable norm has the property that every weakly closed Chebyshev subset, with non-empty weak interior that is norm-wise dense in the subset, is convex.
| Identifer | oai:union.ndltd.org:unt.edu/info:ark/67531/metadc663499 |
| Date | 12 1900 |
| Creators | Svrcek, Frank J. |
| Contributors | Bilyeu, Russell Gene, Mohat, John T., 1924- |
| Publisher | North Texas State University |
| Source Sets | University of North Texas |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
| Format | iii, 69 leaves, Text |
| Rights | Public, Svrcek, Frank J., Copyright, Copyright is held by the author, unless otherwise noted. All rights |
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