Abstract

We study the connections between moduli of asymptotic convexity and smoothness of a Banach space, and the existence of high order differentiable bump functions or equivalent norms on the space. The existence of a high order uniformly differentiable bump function is related to an asymptotically uniformly smooth renorming of power type. On the other hand, the asymptotic uniform convexity of power type is related to the existence of high order rough norms. Finally, we also obtain some applications to the best order smoothness of Nakano sequence spaces

Author information

Contact details are reproduced from the original publication and may be historical.

R. Gonzalo

Dep. de Matemática Aplicada, Facultad de Informática, Universidad Politécnica, Montegancedo / Boadilla del Monte, 28660 Madrid, Spain

rngonzalo@fi.upm.es

J. A. Jaramillo

Dep. de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

jaramil@mat.ucm.es

S. L. Troyanski

Dep. de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo - Murcia, Spain
Permanent Address: Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str. block 8, 1113 Sofia, Bulgaria

stroya@um.es

References

  1. A. D. Alexandrov: Almost everywhere existence of the second order differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Ann. Math. Ser. 6 (1939) 3–35.Google Scholar
  2. R. Aron, J. Globevnik: Analytic functions on c0, Rev. Mat. Univ. Complutense Madr. 2 (1989) 27–33.https://doi.org/10.5209/rev_REMA.1989.v2.18059Google Scholar
  3. O. Blasco, P. Gregori: Type and cotype in vector-valued Nakano sequence spaces, J. Math. Anal. Appl. 264 (2001) 657–672.https://doi.org/10.1006/jmaa.2001.7691Google Scholar
  4. J. M. Borwein, D. Noll: Second order differentiability of convex functions in Banach spaces, Trans. Amer. Math. Soc. 342 (1994) 43–79.https://doi.org/10.2307/2154684Google Scholar
  5. R. Deville, G. Godefroy, V. Zizler: Smooth bump functions and geometry of Banach spaces, Mathematika 40 (1993) 305–321.https://doi.org/10.1112/S0025579300007075Google Scholar
  6. R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Appl. Math. 64, Longman, Harlow (1993).Google Scholar
  7. R. Deville, R. Gonzalo, J. A. Jaramillo: Renormings of Lp (Lq), Math. Proc. Camb. Philos. Soc. 126 (1999) 155–169.https://doi.org/10.1017/S0305004198003119Google Scholar
  8. M. Fabian, D. Preiss, J. H. M. Withfield, V. Zizler: Separating polynomials on Banach spaces, Q. J. Math., Oxf. II. Ser. 40 (1989) 409–422.https://doi.org/10.1093/qmath/40.4.409Google Scholar
  9. M. Fabian, J. H. M. Whitfield, V. Zizler: Norms with locally Lipschitzian derivatives, Israel J. Math. 44 (1983) 262–276.https://doi.org/10.1007/BF02760975Google Scholar
  10. J. Ferrera, J. Gómez, J. G. Llavona: On completion of spaces of weakly continuous functions, Bull. Lond. Math. Soc. 15 (1983) 260–264.https://doi.org/10.1112/blms/15.3.260Google Scholar
  11. T. Figiel, W. B. Johnson: A uniformly convex Banach space which contains no ℓp, Compos. Math. 29 (1974) 179–190.Google Scholar
  12. M. Girardi: The dual of the James tree space is asymptotically uniformly convex, Stud. Math. 147 (2001) 119–130.https://doi.org/10.4064/sm147-2-2Google Scholar
  13. R. Gonzalo: High order smoothness and roughness in Banach Spaces, Q. J. Math. 51 (2000) 75–86.https://doi.org/10.1093/qmathj/50.1.75Google Scholar
  14. R. Gonzalo: Upper and lower estimates in Banach sequence spaces, Commentat. Math. Univ. Carol. 36 (1995) 641–653.Google Scholar
  15. R. Gonzalo, J. A. Jaramillo: Compact polynomials between Banach spaces, Proc. R. Ir. Acad., Sect. A 95 (1995) 213–226.Google Scholar
  16. R. Gonzalo, J. A. Jaramillo: Smoothness and estimates of sequences in Banach spaces, Israel J. Math. 89 (1995) 321–341.https://doi.org/10.1007/BF02808207Google Scholar
  17. R. Gonzalo, J. A. Jaramillo: Separating polynomials on Banach spaces, Extr. Math. 12 (1997) 145–164.Google Scholar
  18. R. Gonzalo, J. A. Jaramillo: High order smoothness in sequence spaces and spreading models, in: Function Spaces, K. Jarosz (ed.), Contemp. Math. 232, AMS, Providence (1999) 151–160.https://doi.org/10.1090/conm/232/03393Google Scholar
  19. P. Hájek, M. Johanis: Isomorphic embeddings and harmonic behaviour of smooth operators, Israel J. Math. 143 (2004) 299–315.https://doi.org/10.1007/BF02803504Google Scholar
  20. R. Haydon: A counterexample to several questions about scattered compact sets, Bull. Lond. Math. Soc. 22 (1990) 261–268.https://doi.org/10.1112/blms/22.3.261Google Scholar
  21. R. Haydon: Trees in renorming theory, Proc. Lond. Math. Soc., III. Ser. 78 (1999) 541–584.https://doi.org/10.1112/S0024611599001768Google Scholar
  22. D. J. Ives: Bump functions and differentiability in Banach spaces, Proc. Amer. Math. Soc. 129 (2001) 3583–3588.https://doi.org/10.1090/S0002-9939-01-06086-5Google Scholar
  23. W. B. Johnson, J. Lindenstrauss, D. Preiss, G. Schechtman: Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. Lond. Math. Soc., III. Ser. 84 (2002) 711–756.https://doi.org/10.1112/S0024611502013400Google Scholar
  24. W. Kaczor, S. Prus: Asymptotical smoothness and its applications, Bull. Aust. Math. Soc. 66 (2002) 405–418.https://doi.org/10.1017/S0004972700040260Google Scholar
  25. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces, I and II, Springer, Berlin (1977) and (1979).https://doi.org/10.1007/978-3-662-35347-9Google Scholar
  26. R. Maleev, B. Zlatanov: Smoothness of Musielak-Orlicz sequence spaces, C. R. Acad. Bulg. Sci. 55 (2002) 11–16.Google Scholar
  27. E. Maluta, S. Prus, M. Szczepanik: On Milman’s moduli for Banach spaces, Abst. Appl. Anal. 6 (2001) 115–129.https://doi.org/10.1155/S1085337501000483Google Scholar
  28. D. McLaughlin, R. Poliquin, J. Vanderweff, V. Zizler: Second order Gâteaux differentiable bump functions and approximations in Banach spaces, Canad. J. Math. 45 (1993) 612–625.https://doi.org/10.4153/CJM-1993-032-9Google Scholar
  29. V. Z. Meshkov: Smoothness properties in Banach spaces, Stud. Math. 63 (1978) 355–369.https://doi.org/10.4064/sm-63-2-111-123Google Scholar
  30. V. D. Milman: Geometric theory of Banach spaces. Part II: Geometry of the unit sphere, Russ. Math. Surv. 26 (1971) 79–163.https://doi.org/10.1070/RM1971v026n06ABEH001273Google Scholar
  31. I. Singer: Bases in Banach Spaces, II, Springer, Berlin (1981).https://doi.org/10.1007/978-3-642-67844-8Google Scholar
  32. C. Stegall: Optimization of functions on certain subsets of Banach spaces, Math. Ann. 236 (1978) 171–176.https://doi.org/10.1007/BF01351389Google Scholar

Suggested citation

R. Gonzalo, J. A. Jaramillo, S. L. Troyanski. “High Order Smoothness and Asymptotic Structure in Banach Spaces.” Journal of Convex Analysis 14 (2007), No. 2, 249–269. https://doi.org/10.68381/jca14018

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.