Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-02-10T11:34:13.860Z Has data issue: false hasContentIssue false
  • English
  • Français

Unconditional bases and martingales in LP(F)

Published online by Cambridge University Press:  24 October 2008

D. J. Aldous
D. J. Aldous
Affiliation:
Statistical Laboratory, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

Here we describe our results and their background: terminology (mostly standard) is denned in Section 2. Throughout, F is a separable Banach space, 1 ≤ p < ∞ and Lp(F) is the space of measurable functions [0,1] → F with P-integrable norms. Given a ‘nice’ property P for Banach spaces, we may formulate the conjecture: Lp(F) satisfies P if and only if both F and Lp (= LP(ℝ)) satisfy P. This conjecture is known to be true for various specific properties, for example the Radon–Nikodym property ((4), section 5·4); reflexivity ((4), corollary 4·1·2); super-refiexivity ((12), proposition 1·2); B-convexity ((14), p. 200); and the properties of not containing copies of c0 (6) and l1 (13). The object of this paper is to demonstrate that the conjecture is false for the property of having an unconditional basis – this answers a question in (4).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 1 , January 1979 , pp. 117 - 123
Copyright
Copyright © Cambridge Philosophical Society 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Billingsley, P.Convergence of probability measures (New York, Wiley, 1968).Google Scholar
(2)Day, M. M.Normed linear spaces, 3rd ed. (Berlin, Springer–Verlag, 1973).CrossRefGoogle Scholar
(3)Diestel, J.Remarks on weak compactness in L 1(μ, X). Glasgow Math. J. 18 (1977), 87–91.CrossRefGoogle Scholar
(4)Diestel, J. and Uhl, J. J.Vector measures (American Math. Society Mathematics Surveys 15, Providence, 1977).CrossRefGoogle Scholar
(5)Garsia, A.Martingale inequalities (Reading, Benjamin, 1973).Google Scholar
(6)Kwapien, S.On Banach spaces containing c o. Studia Math. 52 (1974), 187–188.Google Scholar
(7)Lindenstrauss, J. and Tzafriri, L.Classical Banach spaces (Lecture Notes in Mathematics, no. 338, Berlin, Springer–Verlag, 1973).Google Scholar
(8)Maurey, B. Système de Haar (Seminaire Maurey–Schwartz 1974/5, Ecole Polyteohnique, Paris, 1975).Google Scholar
(9)Maurey, B. and Pisier, G.Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Studio Math. 58 (1976), 45–90.CrossRefGoogle Scholar
(10)Paley, R. E.A remarkable series of orthogonal functions. Proc. London Math. Soc. 34 (1932), 241–264.CrossRefGoogle Scholar
(11)Pisier, G. Un exemple concernant la superréflexivité (Seminaire Maurey–Schwartz 1974/5, Ecole Polytechnique, Paris, 1975).Google Scholar
(12)Pisier, G.Martingales with values in uniformly convex spaces. Israel J. Math. 20 (1975), 326–350.CrossRefGoogle Scholar
(13)Pisier, G. Une propriété de stabilité de la classe des espaces ne contenant l 1. (Preprint.)Google Scholar
(14)Woyczynski, W. A. Geometry and martingales in Banach spaces II. (Preprint.)Google Scholar