Convex sets in Banach spaces and a problem of Rolewicz
Granero, A. ; Jiménez Sevilla, M. ; Moreno, J.
Studia Mathematica, Tome 129 (1998), p. 19-29 / Harvested from The Polish Digital Mathematics Library
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Let Bx be the set of all closed, convex and bounded subsets of a Banach space X equipped with the Hausdorff metric. In the first part of this work we study the density character of Bx and investigate its connections with the geometry of the space, in particular with a property shared by the spaces of Shelah and Kunen. In the second part we are concerned with the problem of Rolewicz, namely the existence of support sets, for the case of spaces C(K).

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:216488 
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Granero, A.; Jiménez Sevilla, M.; Moreno, J. Convex sets in Banach spaces and a problem of Rolewicz. Studia Mathematica, Tome 129 (1998) pp. 19-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv129i1p19bwm/

[00000] [1] J. M. Borwein and J. D. Vanderwerff, Banach spaces that admit support sets, Proc. Amer. Math. Soc. 124 (1996), 751-755. | Zbl 0840.46005

[00001] [2] M. Džamonja and K. Kunen, Properties of the class of measure separable compact spaces, Fund. Math. 147 (1995), 261-277. | Zbl 1068.28502

[00002] [3] C. Finet and G. Godefroy, Biorthogonal systems and big quotient spaces, in: Contemp. Math. 85, Amer. Math. Soc., 1989, 87-110. | Zbl 0684.46016

[00003] [4] J. R. Giles, D. A. Gregory and B. Sims, Characterization of normed linear spaces with Mazur's intersection property, Bull. Austral. Math. Soc. 18 (1978), 471-476. | Zbl 0373.46028

[00004] [5] G. Godefroy, Nicely smooth Banach spaces, in: Functional Analysis Seminar, The University of Texas at Austin, 1984-1985.

[00005] [6] G. Godefroy, Compacts de Rosenthal, Pacific J. Math. 91 (1980), 293-306.

[00006] [7] B. V. Godun and S. L. Troyanski, Renorming Banach spaces with fundamental biorthogonal systems, in: Contemp. Math. 144, Amer. Math. Soc., 1993, 119-126. | Zbl 0801.46008

[00007] [8] M. Jiménez Sevilla and J. P. Moreno, The Mazur intersection property and Asplund spaces, C. R. Acad. Sci. Paris Sér. I 321 (1995), 1219-1223. | Zbl 0836.46011

[00008] [9] M. Jiménez Sevilla and J. P. Moreno, Renorming Banach spaces with the Mazur intersection property, J. Funct. Anal. 144 (1997), 486-504. | Zbl 0898.46008

[00009] [10] M. Jiménez Sevilla and J. P. Moreno, On denseness of certain norms in Banach spaces, Bull. Austral. Math. Soc. 54 (1996), 183-196. | Zbl 0866.46007

[00010] [11] K. Kuratowski, Topology I, Academic Press, New York and London, 1966.

[00011] [12] D. N. Kutzarova, Convex sets containing only support points in Banach spaces with an uncountable minimal system, C. R. Acad. Bulg. Sci. 39 (12) (1986), 13-14. | Zbl 0632.46012

[00012] [13] H. E. Lacey, The Isometric Theory of Classical Banach spaces, Springer, 1974. | Zbl 0285.46024

[00013] [14] A. J. Lazar, Points of support for closed convex sets, Illinois J. Math. 25 (1981), 302-305. | Zbl 0437.46008

[00014] [15] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces, Springer, Berlin, 1979. | Zbl 0403.46022

[00015] [16] S. Mazur, Über schwache Konvergenz in den Räumen Lp, Studia Math. 4 (1933), 128-133. | Zbl 59.1076.01

[00016] [17] V. Montesinos, Solution to a problem of S. Rolewicz, ibid. 81 (1985), 65-69. | Zbl 0526.46016

[00017] [18] S. Negrepontis, Banach spaces and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 1045-1142.

[00018] [19] A. N. Plichko, A Banach space without a fundamental biorthogonal system, Soviet Math. Dokl. 22 (1980), 450-453. | Zbl 0513.46015

[00019] [20] S. Rolewicz, On convex sets containing only points of support, Comment. Math., Tomus specialis in honorem Ladislai Orlicz, I, 1978, 279-281.

[00020] [21] W. Schachermayer, Norm attaining operators and renormings of Banach spaces, Israel J. Math. 44 (1983), 201-212. | Zbl 0542.46013

[00021] [22] A. Sersouri, w-independence in non-separable Banach spaces, in: Contemp. Math. 85, Amer. Math. Soc., 1989, 509-512.

[00022] [23] S. Shelah, Uncountable constructions for B. A., e.c. and Banach spaces, Israel J. Math. 51 (1985), 273-297. | Zbl 0589.03012