Sequential normal compactness versus topological normal compactness in variational analysis

✍ Scribed by Marián Fabian; Boris S. Mordukhovich

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 142 KB
• Volume: 54
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(03)00126-3

✦ Synopsis

We study relationships between two normal compactness properties of sets in Banach spaces that play an essential role in many aspects of variational analysis and its applications, particularly in calculus rules for generalized di erentiation, necessary optimality and suboptimality conditions for optimization problems, etc. Both properties automatically hold in ÿnite-dimensional spaces and reveal principal features of the inÿnite-dimensional variational theory. Similar formulations of these properties involve the weak * convergence of sequences and nets, respectively, containing generalized normal cones in duals to Banach spaces. We prove that these properties agree for a large class of Banach spaces that include weakly compactly generated spaces. We also show that they are always di erent in Banach spaces whose unit dual ball is not weak * sequentially compact. Moreover, the sequential and topological normal compactness properties may not coincide even in non-separable Asplund spaces that admit an equivalent C ∞ -smooth norm.