Complutense University Library

High order smoothness and asymptotic structure in banach spaces

Gonzalo, R. and Jaramillo Aguado, Jesús Ángel and Troyanski, S.L. (2007) High order smoothness and asymptotic structure in banach spaces. Journal of Convex Analysis, 14 (2). pp. 249-269. ISSN 0944-6532

[img] PDF
Restricted to Repository staff only until 2020.

194kB

Official URL: http://www.heldermann-verlag.de/jca/jca14/jca0589_b.pdf

View download statistics for this eprint

==>>> Export to other formats

Abstract

In this paper 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 smoothness of Nakano sequence spaces.

Item Type:Article
Uncontrolled Keywords:Smoothness; bump functions; rough functions; Orlicz spaces
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:16614
References:

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.

R. Aron, J. Globevnik: Analytic functions on c0, Rev. Mat. Univ. Complutense Madr. 2 (1989) 27–33.

O. Blasco, P. Gregori: Type and cotype in vector-valued Nakano sequence spaces, J. Math. Anal. Appl. 264 (2001) 657–672.

J. M. Borwein, D. Noll: Second order differentiability of convex functions in Banach spaces, Trans. Amer. Math. Soc. 342 (1994) 43–79.

R. Deville, G. Godefroy, V. Zizler: Smooth bump functions and geometry of Banach spaces, Mathematika 40 (1993) 305–321.

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).

R. Deville, R. Gonzalo, J. A. Jaramillo: Renormings of Lp(Lq), Math. Proc. Camb. Philos. Soc. 126 (1999) 155–169.

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.

M. Fabian, J. H. M. Whitfield, V. Zizler: Norms with locally Lipschitzian derivatives, Israel J. Math. 44 (1983) 262–276.

J. Ferrera, J. G´omez, J. G. Llavona: On completion of spaces of weakly continuous functions, Bull. Lond. Math. Soc. 15 (1983) 260–264.

T. Figiel, W. B. Johnson: A uniformly convex Banach space which contains no ℓp, Compos. Math. 29 (1974) 179–190.

M. Girardi: The dual of the James tree space is asymptotically uniformly convex, Stud. Math. 147 (2001) 119–130.

R. Gonzalo: High order smoothness and roughness in Banach Spaces, Q. J. Math. 51 (2000) 75–86.

R. Gonzalo: Upper and lower estimates in Banach sequence spaces, Commentat. Math. Univ. Carol. 36 (1995) 641–653.

R. Gonzalo, J. A. Jaramillo: Compact polynomials between Banach spaces, Proc. R. Ir. Acad., Sect. A 95 (1995) 213–226.

R. Gonzalo, J. A. Jaramillo: Smoothness and estimates of sequences in Banach spaces,Israel J. Math. 89 (1995) 321–341.

R. Gonzalo, J. A. Jaramillo: Separating polynomials on Banach spaces, Extr. Math. 12(1997) 145–164.

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.

P. Hájek, M. Johanis: Isomorphic embeddings and harmonic behaviour of smooth operators, Israel J. Math. 143 (2004) 299–315.

R. Haydon: A counterexample to several questions about scattered compact sets, Bull.Lond. Math. Soc. 22 (1990) 261–268.

R. Haydon: Trees in renorming theory, Proc. Lond. Math. Soc., III. Ser. 78 (1999) 541–584.

D. J. Ives: Bump functions and differentiability in Banach spaces, Proc. Amer. Math. Soc.129 (2001) 3583–3588.

W. B. Johnson, J. Lindenstrauss, D. Preiss, G. Schechtman: Almost Fr´echet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. Lond. Math. Soc.,III. Ser. 84 (2002) 711–756.

W. Kaczor, S. Prus: Asymptotical smoothness and its applications, Bull. Aust. Math. Soc. 66 (2002) 405–418.

J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces, I and II, Springer, Berlin (1977) and(1979).

R. Maleev, B. Zlatanov: Smoothness of Musielak-Orlicz sequence spaces, C. R. Acad. Bulg.Sci. 55 (2002) 11–16.

E. Maluta, S. Prus, M. Szczepanik: On Milman’s moduli for Banach spaces, Abst. Appl. Anal. 6 (2001) 115–129.

D. McLaughlin, R. Poliquin, J. Vanderweff, V. Zizler: Second order Gˆateaux differentiable bump functions and approximations in Banach spaces, Canad. J. Math. 45 (1993) 612–625.

V. Z. Meshkov: Smoothness properties in Banach spaces, Stud. Math. 63 (1978) 355–369.

V. D. Milman: Geometric theory of Banach spaces. Part II: Geometry of the unit sphere,Russ. Math. Surv. 26 (1971) 79–163.

I. Singer: Bases in Banach Spaces, II, Springer, Berlin (1981).

C. Stegall: Optimization of functions on certain subsets of Banach spaces, Math. Ann. 236 1978) 171–176.

Deposited On:04 Oct 2012 08:52
Last Modified:07 Feb 2014 09:32

Repository Staff Only: item control page