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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0400-z,
author = {Marek C\'uth and Ond\v rej Kalenda},
title = {Rich families and elementary submodels},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1026-1039},
zbl = {1323.46014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0400-z}
}
Marek Cúth; Ondřej Kalenda. Rich families and elementary submodels. Open Mathematics, Tome 12 (2014) pp. 1026-1039. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0400-z/
[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/S0002-9939-99-05001-7 | Zbl 0937.49009
[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/fm219-3-1 | Zbl 1270.46015
[3] Cúth M., Noncommutative Valdivia compacta, Comment. Math. Univ. Carolin., 2014, 55(1), 53–72 | Zbl 1313.46026
[4] Cúth M., Simultaneous projectional skeletons, J. Math. Anal. Appl., 2014, 411(1), 19–29 http://dx.doi.org/10.1016/j.jmaa.2013.09.020 | Zbl 1308.46033
[5] Cúth M., Rmoutil M., σ-porosity is separably determined, Czechoslovak Math. J., 2013, 63(1), 219–234 http://dx.doi.org/10.1007/s10587-013-0015-3
[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
[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/s11228-013-0256-1 | Zbl 1284.49016
[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.2013.01.008 | Zbl 1272.46009
[9] Garbulińska J., Kubiś W., Remarks on Gurariĭ spaces, Extracta Math., 2011, 26(2), 235–269 | Zbl 1267.46020
[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 | Zbl 1136.46001
[11] Ioffe A.D., On the theory of subdifferentials, Adv. Nonlinear Anal., 2012, 1(1), 47–120 | Zbl 1277.49019
[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/10.1007/978-1-4614-0529-0 | Zbl 1231.46002
[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.2011.07.057 | Zbl 1270.46016
[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.2008.07.006 | Zbl 1166.46008
[15] Kubiś W., Michalewski H., Small Valdivia compact spaces, Topology Appl., 2006, 153(14), 2560–2573 http://dx.doi.org/10.1016/j.topol.2005.09.010 | Zbl 1138.54024
[16] Kunen K., Set Theory, Stud. Logic Found. Math., 102, North-Holland, Amsterdam, 1980
[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 | Zbl 1155.54321
[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 | Zbl 1241.26001
[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/S0002-9939-08-09708-6 | Zbl 1170.46019
[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/s00208-006-0762-7 | Zbl 1112.46015
[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 | Zbl 1245.46033