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On Dentability in Locally Convex Vector Spaces

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Abstract:

The notions of V-dentability, V-s-dentability and V-f-dentability are introduced. It is shown, in particular, that if B is a bounded sequentially complete convex metrizable subset of a locally convex vector space E and V is a neighborhood of zero in E, then the following are equivalent: 1). B is subset V-dentable; 2). B is subset V-s-dentable; 3). B is subset V-f-dentable. It follows from this that for a wide class of locally convex vector spaces E, which strictly contains the class of (BM) spaces (introduced by Elias Saab in 1978), the following is true: every closed bounded subset of E is dentable if and only if every closed bounded subset of E is f-dentable. Also, we get a positive answer to the Saab's question (1978) of whether the subset dentability and the subset s-dentability are the same forthe bounded complete convex metrizable subsets of any l.c.v. space.

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Periodical:
International Journal of Advanced Research in Mathematics (Volume 10)
Pages:
14-19
Citation:
O. Reinov and A. Fahad, "On Dentability in Locally Convex Vector Spaces", International Journal of Advanced Research in Mathematics, Vol. 10, pp. 14-19, 2017
Online since:
September 2017
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References:

[1] E. Saab, On the Radon-Nikodým property in a class of locally convex spaces, Pacific J. Math. 75 (1978) 281-291.

DOI: https://doi.org/10.2140/pjm.1978.75.281

[2] W.J. Davis, R.R. Phelps, The Radon-Nikodým property and dentable sets in Banach spaces, Proc. AMS. 45 (1974) 119-122.

DOI: https://doi.org/10.1090/s0002-9939-1974-0344852-7

[3] O.I. Reinov, On hereditarily dentable sets in Banach spaces, Investigations on linear operators and function theory. Part IX, Zap. Nauchn. Sem. LOMI 92 (1971) 239-243. (in Russian).

[4] R.D. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodým Property, Lecture Notes in Mathematics 993, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, (1983).

DOI: https://doi.org/10.1007/bfb0069321
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