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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 12, Issue 7


Volume 13 (2015)

Rich families and elementary submodels

Marek Cúth
  • Department of Mathematical Analysis, Faculty of Mathematics and Physic, Charles University, Sokolovská 83, Praha 8, 186 75, Czech Republic
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/ Ondřej Kalenda
  • Department of Mathematical Analysis, Faculty of Mathematics and Physic, Charles University, Sokolovská 83, Praha 8, 186 75, Czech Republic
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Published Online: 2014-04-03 | DOI: https://doi.org/10.2478/s11533-013-0400-z


We compare two methods of proving separable reduction theorems in functional analysis — the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system and in spaces of density ℵ1. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality indexed by ranges of its projections.

MSC: 46B26; 03C30

Keywords: Elementary submodel; Separable reduction; Projectional skeleton; Rich family

  • [1] Borwein J.M., Moors W.B., Separable determination of integrability and minimality of the Clarke subdifferential mapping, Proc. Amer. Math. Soc., 2000, 128(1), 215–221 http://dx.doi.org/10.1090/S0002-9939-99-05001-7CrossrefGoogle Scholar

  • [2] Cúth M., Separable reduction theorems by the method of elementary submodels, Fund. Math., 2012, 219(3), 191–222 http://dx.doi.org/10.4064/fm219-3-1CrossrefWeb of ScienceGoogle Scholar

  • [3] Cúth M., Noncommutative Valdivia compacta, Comment. Math. Univ. Carolin., 2014, 55(1), 53–72 Google Scholar

  • [4] Cúth M., Simultaneous projectional skeletons, J. Math. Anal. Appl., 2014, 411(1), 19–29 http://dx.doi.org/10.1016/j.jmaa.2013.09.020CrossrefGoogle Scholar

  • [5] Cúth M., Rmoutil M., σ-porosity is separably determined, Czechoslovak Math. J., 2013, 63(1), 219–234 http://dx.doi.org/10.1007/s10587-013-0015-3CrossrefGoogle Scholar

  • [6] Cúth M., Rmoutil M., Zelený M., On separable determination of σ-P-porous sets in Banach spaces, preprint avaiable at http://arxiv.org/abs/1309.2174 Google Scholar

  • [7] Fabian M., Ioffe A., Separable reduction in the theory of Fréchet subdifferentials, Set-Valued Var. Anal., 2013, 21(4), 661–671 http://dx.doi.org/10.1007/s11228-013-0256-1CrossrefGoogle Scholar

  • [8] Ferrer J., Koszmider P., Kubiś W., Almost disjoint families of countable sets and separable complementation properties, J. Math. Anal. Appl., 2013, 401(2), 939–949 http://dx.doi.org/10.1016/j.jmaa.2013.01.008CrossrefWeb of ScienceGoogle Scholar

  • [9] Garbulińska J., Kubiś W., Remarks on Gurariĭ spaces, Extracta Math., 2011, 26(2), 235–269 Google Scholar

  • [10] Hájek P., Montesinos Santalucía V., Vanderwerff J., Zizler V., Biorthogonal Systems in Banach Spaces, CMS Books Math./Ouvrages Math. SMC, 26, Springer, New York, 2008 Google Scholar

  • [11] Ioffe A.D., On the theory of subdifferentials, Adv. Nonlinear Anal., 2012, 1(1), 47–120 Google Scholar

  • [12] Kąkol J., Kubiś W., López-Pellicer M., Descriptive Topology in Selected Topics of Functional Analysis, Dev. Math., 24, Springer, New York, 2011 http://dx.doi.org/10.1007/978-1-4614-0529-0CrossrefGoogle Scholar

  • [13] Kalenda O.F.K., Kubiś W., Complementation in spaces of continuous functions on compact lines, J. Math. Anal. Appl., 2012, 386(1), 241–257 http://dx.doi.org/10.1016/j.jmaa.2011.07.057CrossrefGoogle Scholar

  • [14] Kubiś W., Banach spaces with projectional skeletons, J. Math. Anal. Appl., 2009, 350(2), 758–776 http://dx.doi.org/10.1016/j.jmaa.2008.07.006CrossrefGoogle Scholar

  • [15] Kubiś W., Michalewski H., Small Valdivia compact spaces, Topology Appl., 2006, 153(14), 2560–2573 http://dx.doi.org/10.1016/j.topol.2005.09.010CrossrefGoogle Scholar

  • [16] Kunen K., Set Theory, Stud. Logic Found. Math., 102, North-Holland, Amsterdam, 1980 Google Scholar

  • [17] Lin P., Moors W.B., Rich families, W-spaces and the product of Baire spaces, Math. Balkanica (N.S.), 2008, 22(1–2), 175–187 Google Scholar

  • [18] Lindenstrauss J., Preiss D., Tišer J., Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Ann. of Math. Stud., 179, Princeton University Press, Princeton, 2012 Google Scholar

  • [19] Moors W.B., Spurný J., On the topology of pointwise convergence on the boundaries of L 1-preduals, Proc. Amer. Math. Soc., 2009, 137(4), 1421–1429 http://dx.doi.org/10.1090/S0002-9939-08-09708-6CrossrefGoogle Scholar

  • [20] Todorcevic S., Biorthogonal systems and quotient spaces via Baire category methods, Math. Ann., 2006, 335(3), 687–715 http://dx.doi.org/10.1007/s00208-006-0762-7CrossrefGoogle Scholar

  • [21] Zajíček L., Generic Fréchet differentiability on Asplund spaces via a.e. strict differentiability on many lines, J. Convex Anal., 2012, 19(1), 23–48 Google Scholar

About the article

Published Online: 2014-04-03

Published in Print: 2014-07-01

Citation Information: Open Mathematics, Volume 12, Issue 7, Pages 1026–1039, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0400-z.

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