In this paper we give an elementary proof of an equivalence theorem of analysis which is often used in optimization theory. The theorem asserts that certain conditions are equivalent to weak convergence in L1. One is the Dunford-Pettis condition concerning absolute integrability. Two others are expressed in terms of Nagumo functions, and can be thought of as growth properties. The original proofs of the various parts of the theorem are scattered in different and specialized mathematical publications. We feel useful to present in this paper a straightforward proof of the various parts in terms of standard Lebesgue integration theory. In particular, the implication (b)⇒(c)∪(d) was proved directly by D. Candeloro and P. Pucci [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 64 (1978), 124-129]. As we pointed out, the theorem can be stated in a more general setting.
An elementary proof of an equivalence theorem relevant in the Theory of Optimization
PUCCI, Patrizia
1985
Abstract
In this paper we give an elementary proof of an equivalence theorem of analysis which is often used in optimization theory. The theorem asserts that certain conditions are equivalent to weak convergence in L1. One is the Dunford-Pettis condition concerning absolute integrability. Two others are expressed in terms of Nagumo functions, and can be thought of as growth properties. The original proofs of the various parts of the theorem are scattered in different and specialized mathematical publications. We feel useful to present in this paper a straightforward proof of the various parts in terms of standard Lebesgue integration theory. In particular, the implication (b)⇒(c)∪(d) was proved directly by D. Candeloro and P. Pucci [Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 64 (1978), 124-129]. As we pointed out, the theorem can be stated in a more general setting.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
