A note on Mackey topologies on Banach spaces

作者：Guirao A J^*; Montesinos V; Zizler V

来源：Journal of Mathematical Analysis and Applications, 2017, 445(1): 944-952.

DOI：10.1016/j.jmaa.2016.08.030

• Banach space
• Mackey topology
• Completeness
• Norming subspace
• Mazur space
• Weak compactness

There is a maybe unexpected connection between three apparently unrelated notions concerning a given w*-dense subspace Y of the dual X* of a Banach space X: (i) The norming character of Y, (ii) the fact that (Y, w*) has the Mazur property, and (iii) the completeness of the Mackey topology mu(X, Y), i.e., the topology on X of the uniform convergence on the family of all absolutely convex w*-compact subsets of Y. To clarify these connections is the purpose of this note. The starting point was a question raised by M. Kunze and W. Arendt and the answer provided by J. Bonet and B. Cascales. We fully characterize mu(X, Y)-completeness or its failure in the case of Banach spaces X with a w*-angelic dual unit ball in particular, separable Banach spaces or, more generally, wealdy compactly generated ones-by using the norming or, alternatively, the Mazur character of Y. We characterize the class of spaces where the original Kunze-Arendt question has always a positive answer. Some other applications are also provided.

• 出版日期2017-1-1