2013/02/25 by Oleg Reinov, Reinov, Oleg, Asfand Fahad +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA
paper · pdf · doi:10.48550/arxiv.1302.6019
5 pages, AMSTeX
arxiv created 2013/02/25 · arxiv updated 2013/02/26
For a locally convex vector space (l.c.v.s.) E and an absolutely convex neighborhood V of zero, a bounded subset A of E is said to be V-dentable (respectively, V-f-dentable) if for any ε>0 there exists an x∈ A so that x∉ co (A∖ (x+εV)) (respectively, so that x∉ co (A∖ (x+εV))). Here, "co" denotes the closure in E of the convex hull of a set. We present a theorem which says that for a wide class of bounded subsets B of locally convex vector spaces the following is true: (V) every subset of B is V-dentable if and only if every subset of B is V-f-dentable. The proof is purely geometrical and independent of any related facts. As a consequence (in the particular case where B is complete convex bounded metrizable subset of a l.c.v.s.), we obtain a positive solution to a 1978-hypothesis of Elias Saab (see p. 290 in "On the Radon-Nikodym property in a class of locally convex spaces", Pacific J. Math. 75, No. 1, 1978, 281-291).