Sequential normal compactness versus topological normal compactness in variational analysis Original Research Article

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 differentiation, necessary optimality and suboptimality conditions for optimization problems, etc. Both properties automatically hold in finite-dimensional spaces and reveal principal features of the infinite-dimensional variational theory. Similar formulations of these properties involve the View the MathML source 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 different in Banach spaces whose unit dual ball is not View the MathML source 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.