Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T03:06:00.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite dimensional characteristics related to superreflexivity of Banach spaces

Published online by Cambridge University Press:  17 April 2009

M. I. Ostrovskii
M. I. Ostrovskii
Affiliation:
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, United States of America, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

One of the important problems of the local theory of Banach Spaces can be stated in the following way. We consider a condition on finite sets in normed spaces that makes sense for a finite set any cardinality. Suppose that the condition is such that to each set satisfying it there corresponds a constant describing “how well” the set satisfies the condition.

The problem is: Suppose that a normed space X has a set of large cardinality satisfying the condition with “poor” constant. Does there exist in X a set of smaller cardinality satisfying the condition with a better constant?

In the paper this problem is studied for conditions associated with one of R.C. James's characterisations of superreflexivity.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 2 , April 2004 , pp. 289 - 295
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Beauzamy, B., Introduction to Banach spaces and their geometry (North-Holland Publishing Company, Amsterdam, 1982).Google Scholar
[2]Bourgain, J., ‘The metrical interpretation of superreflexivity in Banach spaces’, Israel J. Math. 56 (1986), 222–230.CrossRefGoogle Scholar
[3]Burago, D., Burago, Y., and Ivanov, S., A course of metric geometry (American Mathematical Society, Providence, R.I., 2001).CrossRefGoogle Scholar
[4]Boiso, M. Cepedello, ‘Approximation of Lipschitz functions by Δ-convex functions in Banach spaces’, Israel J. Math. 106 (1998), 269–284.CrossRefGoogle Scholar
[5]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, (Longman Scientific & Technical, Harlow, 1993).Google Scholar
[6]van Dulst, D., Reflexive and superreflexive Banach spaces (Mathematisch Centrum, Amsterdam, 1978).Google Scholar
[7]Enflo, P., ‘Banach spaces which can be given an equivalent uniformly convex norm’, Istael J. Math. 13 (1972), 281–288.CrossRefGoogle Scholar
[8]Giesy, D.P., ‘On a convexity condition in normed linear spaces’, Trans. Amer. Math. Soc. 125 (1966), 114–146. Additions and corrections to ‘On a convexity condition in normed liner spaces’, Trans. Amer. Math. Soc. 140 (1969), 511–512.CrossRefGoogle Scholar
[9]Gromov, M., ‘Groups of polynomial growth and expanding maps’, Inst. Hautes Études. Sci. Publ. Math. 53 (1981), 53–73.CrossRefGoogle Scholar
[10]James, R.C., ‘Some self-dual properties of normed linear spaces’, in Symposium on infinite dimensional topology, (Anderson, R.D., Editor) (Princeton University Press, Princeton, New Jersey, 1972), pp. 159–175.CrossRefGoogle Scholar
[11]James, R.C., ‘Super-reflexive spaces with bases’, Pacific J. Math. 41 (1972), 409–419.CrossRefGoogle Scholar
[12]James, R.C. and Schäffer, J.J., ‘Super-reflexivity and the girth of spheres’, Israel J. Math. 11 (1972), 398–404.CrossRefGoogle Scholar
[13]Kadets, M.I., ‘Notes on the gap between subspaces’, (in Russian), Funkts. Anal. Prilozhen. 9 (1975), 73–74. English transl. in Funct. Anal. Appl. 9 (1975), 156–157.Google Scholar
[14]Kadets, M.I., ‘The superreflexivity property of Banach space in terms of the closeness of its finite-dimensional subspaces to Euclidean spaces’, Funct. Anal. Appl. 12 (1978), 142–144.CrossRefGoogle Scholar
[15]Kalton, N.J. and Ostrovskii, M.I., ‘Distances between Banach spaces’ Forum Math. 11 (1999), 17–48.CrossRefGoogle Scholar
[16]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, II, Ergebnisse der Mathematik und ihrer Grenzgebiete 97 (Springer-Verlag, Berlin, 1979).CrossRefGoogle Scholar
[17]Ostrovskii, M.I., ‘Gap properties related characteristics of proximity of Banach spaces’, (in Russian), Teor. Funktsii Funktsional. Anal. i Prilozhen., 42 (1984), 97–107. English transl. in Amer. Math. Soc. Transl. 136 (1987), 109–119.Google Scholar
[18]Ostrovskii, M.I., ‘Banach-Saks properties, injectivity and gaps between subspaces of a Banach space’, (in Russian), Teor. Funktsii, Funktsion. Anal. i Prilozhen. 44 (1985), 69–78. English translation.: J. Soviet Math., 48 (1990), no. 3, 299–306.Google Scholar
[19]Ostrovskii, M.I., ‘Topologies on the set of all subspaces of a Banach space and related questions of Banach space geometry’, Quaestiones Math. 17 (1994), 259–319.CrossRefGoogle Scholar
[20]Ostrovskii, M.I., ‘Classes of Banach spaces stable and unstable with respect to opening’, Quaestiones Math. 19 (1996), 191–210.CrossRefGoogle Scholar
[21]Ostrovskii, M.I., ‘Paths between Banach spaces’, Glasgow Math. J. 44 (2002), 261–273.CrossRefGoogle Scholar
[22]Pisier, G., ‘Martingales with values in uniformly convex spaces’, Israel J. Math. 20 (1975), 326–350.CrossRefGoogle Scholar
[23]Pisier, G. and Xu, Q.. Random series in the real interpolation spaces between the spaces vp, Lecture Notes in Math. 1267 (Springer-Verlag, Berlin, 1987), pp. 185–209.Google Scholar
[24]Schäffer, J.J. and Sundaresan, K., ‘Reflexivity and the girth of spheres’, Math. Ann. 184 (1969/1970), 163–168.CrossRefGoogle Scholar
[25]Wenzel, J., ‘Superreflexivity and J-convexity of Banach spaces’, Acta Math. Univ. Comenian. (N.S.) 66 (1997), 135–147.Google Scholar