Abstract:
In the paper, the problem of preserving the property of approximative compactness under diverse operations is considered. In an arbitrary uniformly convex separable space, we construct an example of two approximatively compact sets whose intersection is not approximatively compact. An example of two linear approximatively compact sets for which the closure of their algebraic sum is not approximatively compact is constructed. In an arbitrary Banach space, we construct two nonlinear approximatively compact sets whose algebraic sum is closed but not approximatively compact. We also prove that any uniformly closed Banach space contains an approximatively compact cavity.
Keywords:
Approximatively compact set, algebraic sum of sets, uniformly closed Banach space, Efimov–Stechkin space.
This publication is cited in the following 5 articles:
Sun L., Sun Yu., Zhang W., Zheng Zh., “On the Sum of Simultaneously Proximinal Sets”, Hacet. J. Math. Stat., 50:3 (2021), 668–677
Kainen P.C., Kurkova V., Vogt A., “Approximative Compactness of Linear Combinations of Characteristic Functions”, J. Approx. Theory, 257 (2020), 105435
Luo Zh., Sun L., Zhang W., “a Remark on the Stability of Approximative Compactness”, J. Funct. space, 2016, 2734947
De la Sen M., “Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings”, J. Appl. Math., 2013, 310106
De la Sen M., “Some Results on Fixed and Best Proximity Points of Multivalued Cyclic Self-Mappings with a Partial Order”, Abstract Appl. Anal., 2013, 968492