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On \(\mathcal T_p\)-locally uniformly rotund norms. (English) Zbl 1288.46008

Summary: Linear topological characterizations of Banach spaces \(E \subset \ell ^{\infty}(\Gamma)\) which admit pointwise locally uniformly rotund norms are obtained. We introduce a new way to construct the norm with families of sliced sets. The topological properties described are related with the theory of generalized metric spaces, in particular with Moore spaces and \(\sigma \)-spaces. A nonlinear transfer is obtained, Question 6.16 in [A. Moltó et al., A nonlinear transfer technique for renorming. Lect. Notes Math. 1951. Berlin: Springer (2009; Zbl 1182.46001)] is answered, and some connections with Kenderov’s school of optimization are presented in this paper.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
46B26 Nonseparable Banach spaces
54E20 Stratifiable spaces, cosmic spaces, etc.
54E30 Moore spaces
54E35 Metric spaces, metrizability
54E40 Special maps on metric spaces

Citations:

Zbl 1182.46001
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[1] Cascales, B., Orihuela, J.: A biased view of topology as a tool in functional analysis. In: Recent Progress in General Topology III, chapt. 3. Springer (2013) · Zbl 1307.54002
[2] Christensen, J.P.R., Kenderov, P.S.: Dense strong continuity of mappings and the Radon-Nykodym property. Math. Scand. 54, 70-78 (1984) · Zbl 0557.46016
[3] Coban, M.M., Kenderov, P.S.: Generic Gateaux differentiability of convex functionals in C(T) and the topological properties of T. In: Proc. of 15th Spring Conference of Union of Bulgarian Math, pp. 141-149. Sljancev, Brjag (1986) · Zbl 1024.46005
[4] Dashiell, F.K., Lindenstrauss, J.: Some examples concerning strictly convex norms on C(K) spaces. Israel J. Math. 16, 329-342 (1973) · Zbl 0281.46011
[5] Deville, R., Godefroy, G., Zizler, V.: Smoothness and renormings in Banach spaces. In: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64. New York (1993) · Zbl 0782.46019
[6] Engelking, R.: General topology. In: PWN—Polish Scientific Publishers, Warsaw (1977). Translated from the Polish by the author, Monografie Matematyczne, Tom 60 [Mathematical Monographs, vol. 60] · Zbl 0373.54002
[7] Fabian, M.: Gateaux differentiability of convex functions and topology. Weak asplund spaces. In: Canadian Mathematical Society Series of Monographs and Advances Texts. Wiley, New York (1997) · Zbl 0883.46011
[8] Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach space theory. The basis for linear and nonlinear analysis. In: CMS Books in Mathematics. Springer, New York (2011) · Zbl 1229.46001
[9] Fabian, M., Montesinos, V., Zizler, V.: Gul’ko, descriptive and Gruenhage compact spaces. RACSAM 104(2), 201-220 (2010) · Zbl 1228.46015
[10] Guirao, A.J.: Pointwise locally uniformly rotund spaces (2012, preprint) · Zbl 0827.54012
[11] Godefroy, G.: Renormings of Banach spaces. In: Handbook of the Geometry of Banach Spaces, vol. I, pp. 781-835. North-Holland, Amsterdam (2001) · Zbl 1009.46003
[12] Gruenhage, G.: Generalized metric spaces. In: Handbook of Set-Theoretic Topology, pp. 423-501. North-Holland, Amsterdam (1984) · Zbl 0555.54015
[13] Hájek, P., Montesinos Santalucía, V., Vanderwerff, J., Zizler, V.: Biorthogonal systems in Banach spaces. In: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 26. Springer, New York (2008) · Zbl 1136.46001
[14] Kenderov, P.S.: Monotone operators in Asplund spaces. C. R. Acad. Bulg. Sci. 30, 963-964 (1977) · Zbl 0377.47036
[15] Kenderov, P.S.: Most optimization problems have unique solution. In: International Series in Numerical Mathematics, vol. 72, pp. 203-215. Basel, Birkhauser (1984) · Zbl 0541.49006
[16] Kenderov, P.S., Moors, W.B.: Fragmentability of Banach spaces. C. R. Acad. Bulg. Sci. 49, 9-12 (1996) · Zbl 0891.46008
[17] Kenderov, P.S., Moors, W.B.: Fragmentability and sigma-fragmentability of Banach spaces. J. Lond. Math. Soc. (2), 60(1), 203-223 (1999) · Zbl 0953.46004
[18] Kenderov, P.S., Orihuela, J.: A generic factorization theorem. Mathematika 42, 56-66 (1995) · Zbl 0827.54012
[19] Kenderov, P.S., Revalski, J.P.: The Banach-Mazur game and generic existence of solutions to optimization problems. Proc. Am. Math. Soc. 118, 911-917 (1993) · Zbl 0794.90089
[20] Haydon, R.: Trees in renorming theory. Proc. Lond. Math. Soc. 78(3), 541-584 (1999) · Zbl 1036.46003
[21] Haydon, R.: Locally uniformly convex norms in Banach spaces and their duals. J. Funct. Anal. 254(8), 2023-2039 (2008) · Zbl 1158.46005
[22] Haydon, R., Moltó, A., Orihuela, J.: Spaces of functions with countably many discontinuities. Israel J. Math. 158, 19-39 (2007) · Zbl 1133.54010
[23] Mercourakis, S.: On weakly countably determined Banach spaces. Trans. Am. Math. Soc. 300, 307-327 (1987) · Zbl 0621.46018
[24] Moltó, A., Orihuela, J., Troyanski, S.: Locally uniformly rotund renorming and fragmentability. Proc. Lond. Math. Soc. (3) 75(3), 619-640 (1997) · Zbl 0909.46011
[25] Moltó, A., Orihuela, J., Troyanski, S., Valdivia, M.: On weakly locally uniformly rotund Banach spaces. J. Funct. Anal. 163(2), 252-271 (1999) · Zbl 0927.46010
[26] Moltó, A.; Orihuela, J.; Troyanski, S.; Valdivia, M., A nonlinear transfer technique for renorming (2009), New York · Zbl 1182.46001
[27] Moors, W.B.: Some more results concerning weak Asplund spaces. Abstr. Appl. Anal. 2005(3), 307-318 (2005) · Zbl 1107.46013
[28] Oncina, L., Raja, M.: Descriptive compact spaces and renorming. Stud. Math. 165(1), 39-52 (2004) · Zbl 1101.46014
[29] Orihuela, J., Troyanski, S.: Devilles’s master lemma and stone discretness in renorming theory. J. Convex Anal. 16(4), 959-972 (2009) · Zbl 1192.46009
[30] Orihuela, J., Troyanski, S.: LUR renormings through Deville’s master lemma. RACSAM (2009) 103, 75-85 (2009) · Zbl 1192.46008
[31] Preiss, D., Phelps, R.R., Namioka, I.: Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings. Israel J. Math. 72, 257-279 (1990) · Zbl 0757.46028
[32] Ribarska, N.K.: Internal characterization of fragmentable spaces. Mathematika 34, 243-257 (1987) · Zbl 0645.46017
[33] Raja, M.: Locally uniformly rotund norms. Mathematika 46(2), 343-358 (1999) · Zbl 1031.46022
[34] Raja, M.: Weak* locally uniformly rotund norms and descriptive compact spaces. J. Funct. Anal. 197(1), 1-13 (2003) · Zbl 1024.46005
[35] Smith, R.J., Troyanski, S.: Renormings of C(K) spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 104(2), 375-412 (2010) · Zbl 1227.46012
[36] Zizler, V.: Nonseparable Banach spaces. In: Handbook of the Geometry of Banach Spaces, vol. 2., pp. 1745-1816. Elsevier Science B.V. (2003) · Zbl 1041.46009
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