Linearly and directionally bounded weak-star closed sets and the AFPP
- 3 Downloads
Linearly bounded and directionally bounded closed convex sets play a very relevant role in metric fixed point theory [12, 14, 16]. In reflexive spaces both collections of sets are identical and this fact characterizes the reflexivity of the space [14, 16]. Since closed convex sets are weakstar closed in reflexive spaces, it is natural to ask about the relationship between linearly bounded and directionally bounded sets in the case of a non-reflexive dual space if we assume, in addition, that the set is weak-star closed. We will show two divergent answers to this question depending on certain topological and isometric properties of the underlying dual Banach space.
Unable to display preview. Download preview PDF.
- J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92, Springer, Berlin–New York, 1977.Google Scholar
- E. Matoušková and S. Reich, Reflexivity and approximate fixed points, Studia Mathematica 159 (2003), no. 3, 403-415.Google Scholar