By Klaus-Dieter Bierstedt
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Extra resources for Functional Analysis: Surveys and Recent Results II
Rosenthal was able to show that the second of these conditions is always satisfied. 26 R. (Rosentha1 (19)) Every infinite-dimensional injective Banach space has a subspace isomorphic to £00' Hence, for an infinite-dimensional injective Banach space X the following are equivalent: (a) X is isomorphic to £00 ; (b) X embeds in £00 (c) ba11X* is weak* separable. Thus, 'reasonably small' injectives are isomorphic to £00' We can now ask what injectives can be shown to be isomorphic to £oo(A) if we allow uncountable sets A .
In the meantime we have proved 2B Theorem. (Pefczyfiski (16)) The Banach spaces Loo and £00 are isomorphic. We have already noted that £1 and Ll are not isomorphic because of the existence of Hilbert subspaces of Ll In fact there is an even more striking difference between these two spaces. One can observe that all the £p spaces NON-SEPARABLE BANACH SPACES 21 as well as Lp for l
Ct. Anal. 17 (1974),1(:'1-'1'73. 111\ L. Azimov and J. Ellis: Facial decomposition of linear compact 0implexe:~ and separation of functions on cones, Pacific J. Math. 34 (1970), 301-309. J. Bliedtner: In Proc. of the second Paderborn conference on funct. anal, edited by Bierstedt and Fuchssteiner, this vol. P. Bourgin: Barycenters of mea:;ures on certain non-compact convex 2,et~;; Trans MIS 154 (1971),323-340. P. A. Edgar: Non-compact simplexe:3 in Banach ;3paces with the Radon-Nikodym property; J.