A Banach space X is an M-ideal in its bidual if the relation
||f + w|| = ||f|| + ||w||
holds for every f in X* and every w in X ⊥.
The class of the Banach spaces which are M-ideals in their biduals, in short, the class of M-embedded spaces, has been carefully investigated, in particular by A. Lima, G. Godefroy and the West Berlin School. The spaces c0(I) -I any set- equipped with their canonical norm belong to this class, which also contains e.g. certain spaces K(E,F) of compact operators between reflexive spaces (see [7]). This class has very nice properties; for instance, these are Weakly Compactly Generated (W.C.G.) Asplund spaces [2; Th. 3], have the property (v) [5; Th. 1] and (u) [4; Main Th.] of Pelczynski and satisfy, among other isometric properties, that every isometric isomorphism of X** is the bitranspose of an isometric isomorphism of X [6; Prop. 4.2]. The purpose of this work is to show that these properties are also true in a wider class of Banach spaces.
@article{urn:eudml:doc:39875,
title = {On Banach spaces which are M-ideals in their biduals.},
journal = {Extracta Mathematicae},
volume = {5},
year = {1990},
pages = {74-76},
zbl = {1231.46007},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39875}
}
Cabello Piñar, Juan Carlos. On Banach spaces which are M-ideals in their biduals.. Extracta Mathematicae, Tome 5 (1990) pp. 74-76. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39875/