RT Journal Article
T1 On some subsets of L 1 (μ,E)
A1 Bombal Gordón, Fernando
AB The paper starts with the following remark: One of the most common methods used in the literature to introduce new properties in a Banach space E  consists in establishing some nontrivial relationships between different classes of subsets of E . Moving on from this, the author considers the classes of bounded, weakly relatively compact, weakly conditionally compact, norm relatively compact, Dunford-Pettis, and (V* )  subsets of L 1 (μ,E)  (in symbols: B,W,WC,K,DP,V* , respectively) and investigates their nature and the consequences of the possible coincidence of two of them in terms of properties of the space L 1 (μ,E) . He observes that the following necessary condition is true. Proposition II.1: Let H  be any of the classes K,W,WC,DP  and V*  . If M H(L 1 (μ,E))  then: (a) M  is bounded; (b) M  is uniformly integrable; (c) for every measurable set A , M(A)={∫ A fdμ , f K}  is in H(E) . Then he gives the following definition: A subset M  of L 1 (μ,E)  satisfying conditions (a) to (c) of Proposition II.1 is called a μH -set; a Banach space E  is said to have property P(μ,H)  if every μH -set belongs to H(L 1 (μ,E)) . Then he gives necessary and sufficient conditions for a Banach space E  to have property P(μ,V*),P(μ,WC)  and P(μ,DP)
PB Springer Verlag
SN 0011-4642
YR 1991
FD 1991
LK https://hdl.handle.net/20.500.14352/57912
UL https://hdl.handle.net/20.500.14352/57912
LA eng
NO DGICYT
DS Docta Complutense
RD 9 abr 2025