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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some converses of the strong
separation theorem


Authors: Hwa-Long Gau and Ngai-Ching Wong
Journal: Proc. Amer. Math. Soc. 124 (1996), 2443-2449
MSC (1991): Primary 46A03, 46A25, 46B10
MathSciNet review: 1327015
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Abstract: A convex subset $B$ of a real locally convex space $X$ is said to have the separation property if it can be separated from every closed convex subset $A$ of $X$, which is disjoint from $B$, by a closed hyperplane. The strong separation theorem says that if $B$ is weakly compact, then it has the separation property. In this paper, we present two versions of the converse and discuss an application of them. For example, we prove that a normed space is reflexive if and only if its closed unit ball has the separation property. Results in this paper can be considered as supplements of the famous theorem of James.


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Additional Information

Hwa-Long Gau
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan, Republic of China
Address at time of publication: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan, Republic of China
Email: u8222807@cc.nctu.edu.tw

Ngai-Ching Wong
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan, Republic of China
Email: wong@math.nsysu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03343-6
PII: S 0002-9939(96)03343-6
Received by editor(s): October 18, 1994
Received by editor(s) in revised form: February 22, 1995
Additional Notes: This research is partially supported by National Science Council of Taiwan, R.O.C
Dedicated: To the memory of Yau-Chuen Wong (1935.10.2 – 1994.11.7)
Communicated by: Dale Alspach
Article copyright: © Copyright 1996 American Mathematical Society