Publication: Asymptotic structure, l(p)-estimates of sequences, and compactness of multilinear mappings
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2009
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Academic Press- Elsevier Science
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We relate the moduli of asymptotic uniform smoothness and convexity of a Banach space with the existence of upper and lower l(p)-estimates of sequences in the space. To this end, we introduce two properties which are related to the (m(p))-property defined by Kalton and Werner. In this way we obtain a connection between the moduli of asymptotic uniform smoothness and convexity, and compactness or weak-sequential continuity of multilinear mappings. Finally, we give some applications to the existence of analytic and asymptotically flat norms on a Banach space.
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R. Alencar, K. Floret, Weak-strong continuity of multilinear mappings and the Pełczy´nski–Pitt theorem, J. Math. Anal. Appl. 206 (1997) 532–546.
R. Aron, C. Hervés, M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983) 189–204.
J.M.F. Castillo, F. Sánchez,Weakly p-compact, p-Banach-Saks and super-reflexive Banach spaces, J. Math. Anal. Appl. 185 (1994) 256–261.
J.M.F. Castillo, R. García, R. Gonzalo, Banach spaces in which all multilinear forms are weakly sequentially continuous, Studia Math. 136 (1999) 121–145.
M. Cepedello, P. Hájek, Analytic approximation of uniformly continuous functions on real Banach spaces, J. Math. Anal. Appl. 256 (2001) 80–98.
R. Deville, V.H. Fonf, P. Hájek, Analytic and polyhedral approximation of convex bodies in separable polyhedral spaces, Israel J. Math. 105 (1998) 139–154.
R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Longman Scientific & Technical, Harlow, 1993.
V. Dimant, I. Zalduendo, Bases in spaces of multilinear forms over Banach spaces, J. Math. Anal. Appl. 200 (1996) 548–566.
S. Delpech, Appoximations höldériennes de fonctions entre espaces d’Orlicz. Modules asymptotiques uniformes, Doctoral Thesis, Université de Bordeaux, 2005.
N. Dew, Asymptotic structure of Banach spaces, Doctoral Thesis, University of Oxford, 2002.
G. Godefroy, The Banach space c0, Extracta Math. 16 (2001) 1–26.
G. Godefroy, N. Kalton, G. Lancien, Subspaces of c0(N) and Lipschitz isomorphisms, Geom. Funct. Anal. 10 (2000) 798–820.
V. Fonf, A property of Lindenstrauss–Phelps spaces, Funktsional. Anal. i Prilozhen. 13 (1979) 79–80 (in Russian).
R. Gonzalo, Suavidad y polinomios en espacios de Banach, Doctoral Thesis, Universidad Complutense de Madrid, 1994.
R. Gonzalo, Upper and lower estimates in Banach sequence spaces, Comment. Math. Univ. Carolin. 36 (1995) 641–653.
R. Gonzalo, Polynomial norms in Banach spaces, preprint.
R. Gonzalo, J.A. Jaramillo, Compact polynomials between Banach spaces, Proc. R. Ir. Acad. 95 (1995) 213–226.
R. Gonzalo, J.A. Jaramillo, High order smoothness in sequence spaces and spreading models, Contemp. Math. 232 (1999) 151–160.
R. Gonzalo, J.A. Jaramillo, S. Troyanski, High order smoothness and asymptotic structure in Banach spaces, J. Convex Anal. 14 (2007) 249–269.
P. Hájek, Smooth norms that depend locally on finitely many coordinates, Proc. Amer. Math. Soc. 123 (1995) 3817–3821.
P. Hájek, S. Troyanski, Analytic norms in Orlicz spaces, Proc. Amer. Math. Soc. 129 (2001) 713–717.
W.B. Johnson, E. Odell, Subspaces of Lp which embed into _p, Compos. Math. 28 (1974) 37–49.
[23] W.B. Johnson, J. Lindenstrauss, D. Preiss, G. Schechtman, Almost Fréchet differentiability of Lipschitz mappings between infinite dimensional Banach spaces, Proc. London Math. Soc. 84 (2002) 711–746.
N. Kalton, D. Werner, Property (M), M-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995) 137–178.
H. Knaust, E. Odell, Weakly null sequences with upper _p-estimates, in: Lecture Notes in Math., vol. 1470, Springer-Verlag, Berlin, 1991, pp. 137–178.
D.H. Leung, Some isomorphically polyhedral Orlicz sequence spaces, Israel J. Math. 87 (1994) 117–128.
J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces. I, Springer-Verlag, Berlin, 1977.
J.E. Littlewood, On bounded bilinear forms in an infinite number of vaiables, Q. J. Math. (Oxford) 1 (1930) 164–174.
V.D. Milman, Geometric theory of Banach spaces. Part II, Geometry of the unit sphere, Uspekhi Mat. Nauk 26 (1971) 73–149 (in Russian), English transl. in Russian Math. Surveys 26 (1971) 79–163.
A. Pełczy´nski, A property of multilinear operations, Studia Math. 16 (1957) 173–182.
S. Prus, On infinite dimensional uniform smoothness, Comment. Math. Univ. Carolin. 40 (1999) 97–105