# Article

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Keywords:
Asplund generated space; continuous image of Radon-Nikodym compact; totally disconnected compact; adequate compact; Eberlein compact
Summary:
A family of compact spaces containing continuous images of Radon-Nikod'ym compacta is introduced and studied. A family of Banach spaces containing subspaces of Asplund generated (i.e., GSG) spaces is introduced and studied. Further, for a continuous image of a Radon-Nikod'ym compact $K$ we prove: If $K$ is totally disconnected, then it is Radon-Nikod'ym compact. If $K$ is adequate, then it is even Eberlein compact.
References:
[A1] Argyros S.: Weakly Lindelöf determined Banach spaces not containing $\ell_1$. preprint.
[A2] Argyros S.: Private communication.
[BRW] Benyamini Y., Rudin M.E., Wage M.: Continuous images of weakly compact subsets of Banach spaces. Pacific J. Math. 70 (1977), 309-324. MR 0625889 | Zbl 0374.46011
[DS] Dunford N., Schwartz J.T.: Linear Operators, Part I. Interscience Publ., New York, 1958. Zbl 0635.47001
[F] Fabian M.: Gâteaux Differentiability of Convex Functions and Topology - Weak Asplund Spaces. John Wiley & Sons, Interscience, New York, 1997. MR 1461271 | Zbl 0883.46011
[Gr] Grothendieck A.: Produits tensoriels et espaces nucleaires. Memoirs Amer. Math. Soc., No. 16, 1955. MR 0075539
[H1] Heisler M.: Singlevaluedness of monotone operators on subspaces of GSG spaces. Comment. Math. Univ. Carolinae 37 (1996), 255-261. MR 1399000 | Zbl 0849.47025
[H2] Heisler M.: Some aspects of differentiability and geometry on Banach spaces. PhD. Thesis, Prague, 1996.
[N1] Namioka I.: Eberlein and Radon-Nikodým compact spaces. Lecture Notes at University College, London, 1985. MR 0963600
[N2] Namioka I.: Radon-Nikodým compact spaces and fragmentability. Mathematika 34 (1987), 258-281. MR 0933504 | Zbl 0654.46017
[NP] Namioka I., Phelps R.R.: Banach spaces which are Asplund spaces. Duke Math. J. 42 (1975), 735-750. MR 0390721 | Zbl 0332.46013
[OSV] Orihuela J., Schachermayer W., Valdivia M.: Every Radon-Nikodým Corson compact is Eberlein compact. Studia Math. 98 (1991), 157-174. MR 1100920
[Ph] Phelps R.R.: Convex functions, monotone operators and differentiability. Lect. Notes Math., No. 1364, 2nd Edition, Springer Verlag, Berlin, 1993. MR 1238715 | Zbl 0921.46039
[Ro] Rosenthal H.: The heredity problem for weakly compactly generated Banach spaces. Comp. Math. 28 (1974), 83-111. MR 0417762 | Zbl 0298.46013
[St1] Stegall Ch.: The Radon-Nikodým property in conjugate Banach spaces II. Trans. Amer. Math. Soc. 264 (1981), 507-519. MR 0603779 | Zbl 0475.46016
[St2] Stegall Ch.: More facts about conjugate Banach spaces with the Radon-Nikodým property II. Acta Univ. Carolinae - Math. et Phys. 32 (1991), 47-54. MR 1146766 | Zbl 0773.46008
[T] Talagrand M.: Espaces de Banach faiblement $K$-analytiques. Annals of Math. 110 (1978), 407-438. MR 0554378

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