Dual Kadec–Klee property and fixed points

A Banach space X is said to have the dual Kadec–Klee property iff on the unit sphere of the dual space X ⁎ the weak*-topology coincides with the norm-topology. A. Amini-Harandi and M. Fakhar extended a theorem of B. Ricceri concerning the Kadec–Klee property by showing that if X has the dual Kadec–K...
Ausführliche Beschreibung

A Banach space X is said to have the dual Kadec–Klee property iff on the unit sphere of the dual space X ⁎ the weak*-topology coincides with the norm-topology. A. Amini-Harandi and M. Fakhar extended a theorem of B. Ricceri concerning the Kadec–Klee property by showing that if X has the dual Kadec–Klee property then for every compact mapping φ : B X ⁎ → X ∖ { 0 } there exists some f in the unit sphere of X ⁎ such that 〈 f , φ ( f ) 〉 = ‖ φ ( f ) ‖ and conjectured that this yields a necessary and sufficient condition for X to have the dual Kadec–Klee property. We prove here that it is almost the case: we introduce a weakening of the dual Kadec–Klee property, here called the NAKK* property, and show that this property is equivalent to the above property about compact mappings from B X ⁎ to X ∖ { 0 } . Ausführliche Beschreibung