Some converses of the strong separation theorem
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- by Hwa-Long Gau and Ngai-Ching Wong PDF
- Proc. Amer. Math. Soc. 124 (1996), 2443-2449 Request permission
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.References
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
- Robert C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140. MR 165344, DOI 10.1090/S0002-9947-1964-0165344-2
- Robert C. James, Weak compactness and separation, Canadian J. Math. 16 (1964), 204–206. MR 157219, DOI 10.4153/CJM-1964-020-x
- Robert C. James, Weak compactness and reflexivity, Israel J. Math. 2 (1964), 101–119. MR 176310, DOI 10.1007/BF02759950
- Robert C. James, A counterexample for a $\textrm {sup}$ theorem in normed spaces, Israel J. Math. 9 (1971), 511–512. MR 279565, DOI 10.1007/BF02771466
- Robert C. James, Reflexivity and the sup of linear functionals, Israel J. Math. 13 (1972), 289–300 (1973). MR 338742, DOI 10.1007/BF02762803
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- J. D. Pryce, Weak compactness in locally convex spaces, Proc. Amer. Math. Soc. 17 (1966), 148–155. MR 190695, DOI 10.1090/S0002-9939-1966-0190695-2
- Charles Hopkins, An extension of a theorem of Remak, Ann. of Math. (2) 40 (1939), 636–638. MR 33, DOI 10.2307/1968948
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
- 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
- Communicated by: Dale Alspach
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2443-2449
- MSC (1991): Primary 46A03, 46A25, 46B10
- DOI: https://doi.org/10.1090/S0002-9939-96-03343-6
- MathSciNet review: 1327015
Dedicated: To the memory of Yau-Chuen Wong (1935.10.2 – 1994.11.7)