Almost antiproximinal sets in $L_1$

Author V. A. Kholomenyuk, V. V. Mykhaylyuk

Jurii Fedkovych Chernivtsy National University

Abstract We introduce a notion of almost antiproximinality of sets in the space $L_1$ which is a weaken- ing of the notion of antiproximinality. Also we investigate properties of almost antiproximinal sets and establish a method of construction of almost antiproximinal sets.
Keywords almost antiproximinal set; space $L_1$
Reference 1. Edelstein M., Thompson A.C. Some results on nearest points and support properties of convex sets in c0// Pacific J. Math. 1972. V.40. P. 553560.

2. Bishop E., Phelps R.R. A proof that every Banach space is subreflexive// Bull. Amer. Math. Soc. 1961. V.67. P. 9798.

3. Bishop E., Phelps R.R. Support functionals of convex sets// Proc.Simposia in Pure Math. (Convexity) Amer. Math. Soc. 1963. V.7. P. 2735.

4. Klee V. Remarks on nearest points in normed linear spaces// Proc. Colloquium on Convexity, Copenhagen. 1965. P. 168176.

5. Cobzas S. Antiproximinal sets in the spaces $c_0$ and $c$// Math. Notes. 1975. V.17. P. 449457.

6. Fonf V.P. On antiproximinal sets in spaces of continuous functions on compacta// Mat. Zametki. 1983. V.33, 4. P. 549558. (in Russian)

7. Balaganskii V.S. Antiproximinal sets in the space of continuous functions// Math. Notes. 1996. V.60, 5. P. 485494.

8. Kantorovich L.V., Akilov G.P. Functional analysis. Moscow: Nauka, 1984. 752p. (in Russian)

9. Schaefer H. Topological vector spaces. Moscow: Mir, 1971. 359p. (in Russian)

10. Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. Moscow: Nauka, 1976. 544p. (in Russian)

11. Natanson I.P. Theory of functions of real variable. Moscow: Nauka, 1974. 480p. (in Russian)

12. Martinez-Abejon A., Odell E., Popov M.M. Some open problems on the classical function space $L_1$// Mat. Stud. 2005. V.24, 2. P. 173191.

Pages 172-180
Volume 35
Issue 2
Year 2011
Journal Matematychni Studii
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