Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

$ C^k$-smooth approximations of LUR norms


Authors: Petr Hájek and Antonín Procházka
Journal: Trans. Amer. Math. Soc. 366 (2014), 1973-1992
MSC (2010): Primary 46B20, 46B03, 46E15
DOI: https://doi.org/10.1090/S0002-9947-2013-05899-0
Published electronically: December 13, 2013
MathSciNet review: 3152719
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a WCG Banach space admitting a $ C^{k}$-smooth norm where $ k \in \mathbb{N} \cup \left \{\infty \right \}$. Then $ X$ admits an equivalent norm which is simultaneously, $ C^1$-smooth, LUR, and the limit of a sequence of $ C^{k}$-smooth norms. If $ X=C([0,\alpha ])$, where $ \alpha $ is any ordinal, then the same conclusion holds true with $ k=\infty $.


References [Enhancements On Off] (What's this?)

  • [1] Edgar Asplund, Averaged norms, Israel J. Math. 5 (1967), 227-233. MR 0222610 (36 #5660)
  • [2] Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow, 1993. MR 1211634 (94d:46012)
  • [3] Robert Deville, Vladimir Fonf, and Petr Hájek, Analytic and $ C^k$ approximations of norms in separable Banach spaces, Studia Math. 120 (1996), no. 1, 61-74. MR 1398174 (97h:46012)
  • [4] Robert Deville, Vladimir Fonf, and Petr Hájek, Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces, Israel J. Math. 105 (1998), 139-154. MR 1639743 (99h:46006), https://doi.org/10.1007/BF02780326
  • [5] Pavel Drábek and Jaroslav Milota, Methods of nonlinear analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007. Applications to differential equations. MR 2323436 (2008i:47134)
  • [6] Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8, Springer-Verlag, New York, 2001. MR 1831176 (2002f:46001)
  • [7] Marián Fabian, Petr Hájek, and Václav Zizler, A note on lattice renormings, Comment. Math. Univ. Carolin. 38 (1997), no. 2, 263-272. MR 1455493 (98e:46008)
  • [8] Marian Fabian, Vicente Montesinos, and Václav Zizler, Smoothness in Banach spaces. Selected problems, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006), no. 1-2, 101-125 (English, with English and Spanish summaries). MR 2267403 (2007g:46026)
  • [9] M. Fabián, J. H. M. Whitfield, and V. Zizler, Norms with locally Lipschitzian derivatives, Israel J. Math. 44 (1983), no. 3, 262-276. MR 693663 (84i:46028), https://doi.org/10.1007/BF02760975
  • [10] Petr Hájek, On convex functions in $ c_0(\omega _1)$, Collect. Math. 47 (1996), no. 2, 111-115. MR 1402064 (97h:46069)
  • [11] Petr Hájek, Vicente Montesinos Santalucía, Jon Vanderwerff, and Václav Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 26, Springer, New York, 2008. MR 2359536 (2008k:46002)
  • [12] P. Hájek, A. Procházka, Smooth version of Deville's master lemma, preprint (2009).
  • [13] P. Hájek, J. Talponen, Smooth approximations of norms in separable Banach spaces, arXiv:1105.6046v1, 2011.
  • [14] Richard Haydon, Smooth functions and partitions of unity on certain Banach spaces, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 188, 455-468. MR 1460234 (2000c:46080), https://doi.org/10.1093/qmath/47.4.455
  • [15] D. McLaughlin, R. Poliquin, J. Vanderwerff, and V. Zizler, Second-order Gateaux differentiable bump functions and approximations in Banach spaces, Canad. J. Math. 45 (1993), no. 3, 612-625. MR 1222519 (94g:46025), https://doi.org/10.4153/CJM-1993-032-9
  • [16] J. Pechanec, J. H. M. Whitfield, and V. Zizler, Norms locally dependent on finitely many coordinates, An. Acad. Brasil. Ciênc. 53 (1981), no. 3, 415-417. MR 663236 (83h:46025)
  • [17] Michel Talagrand, Renormages de quelques $ {\mathcal {C}}(K)$, Israel J. Math. 54 (1986), no. 3, 327-334 (French, with English summary). MR 853457 (88d:46041), https://doi.org/10.1007/BF02764961
  • [18] S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1970/71), 173-180. MR 0306873 (46 #5995)
  • [19] Václav Zizler, Nonseparable Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1743-1816. MR 1999608 (2004g:46030), https://doi.org/10.1016/S1874-5849(03)80048-7

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46B20, 46B03, 46E15

Retrieve articles in all journals with MSC (2010): 46B20, 46B03, 46E15


Additional Information

Petr Hájek
Affiliation: Mathematical Institute, Czech Academy of Science, Žitná 25, 115 67 Praha 1, Czech Republic – and – Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 166 27 Prague 6, Czech Republic
Email: hajek@math.cas.cz

Antonín Procházka
Affiliation: Laboratoire de Mathématiques UMR 6623, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France
Email: antonin.prochazka@univ-fcomte.fr

DOI: https://doi.org/10.1090/S0002-9947-2013-05899-0
Keywords: LUR, smoothness, higher order smoothness, renorming
Received by editor(s): January 22, 2009
Received by editor(s) in revised form: April 4, 2011, May 3, 2012, and June 19, 2012
Published electronically: December 13, 2013
Additional Notes: This work was supported by grants GA CR Grant P201/11/0345, RVO: 67985840, and PHC Barrande 2012 26516YG
Article copyright: © Copyright 2013 by the authors

American Mathematical Society