Descriptive properties of elements of biduals of Banach spaces

Pavel Ludvík ¹   ; Jiří Spurný ²

¹ Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University Sokolovská 83 186 75 Praha 8, Czech Republic
² Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University Sokolovská 83, 186 75 Praha 8, Czech Republic

Studia Mathematica, Tome 209 (2012) no. 1, pp. 71-99

Résumé

If $E$ is a Banach space, any element $x^{**}$ in its bidual $E^{**}$ is an affine function on the dual unit ball $B_{E^*}$ that might possess a variety of descriptive properties with respect to the weak$^*$ topology. We prove several results showing that descriptive properties of $x^{**}$ are quite often determined by the behaviour of $x^{**}$ on the set of extreme points of $B_{E^*}$, generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire classes of $L_1$-preduals which were introduced by S. A. Argyros, G. Godefroy and H. P. Rosenthal (2003). Also, several examples witnessing natural limits of our positive results are presented.

Détail

DOI : 10.4064/sm209-1-6

Keywords: banach space element ** its bidual ** affine function dual unit ball * might possess variety descriptive properties respect weak * topology prove several results showing descriptive properties ** quite often determined behaviour ** set extreme points * generalizing results nbsp saint raymond nbsp jellett prove result relation between baire classes intrinsic baire classes preduals which introduced nbsp nbsp argyros nbsp godefroy nbsp nbsp rosenthal several examples witnessing natural limits positive results presented