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Convexity conditions for non-locally convex lattices

Published online by Cambridge University Press:  18 May 2009

N. J. Kalton
N. J. Kalton
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211
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First we recall that a (real) quasi-Banach space X is a complete metrizable real vector space whose topology is given by a quasi-norm satisfying

where C is some constant independent of x1 and x2. X is said to be p-normable (or topologically p-convex), where 0 < p ≤ l, if for some constant B we have

for any x1, …, xn, є X. A theorem of Aolci and Rolewicz (see [18]) asserts that if in C = 21/p-1, then X is p-normable. We can then equivalently re-norm X so that in (1.4) B = 1.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 2 , July 1984 , pp. 141 - 152
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

1.Bennett, G., An extension of the Riesz-Thorin theorem in Banach spaces of analytic functions. Lecture Notes in Mathematics No 604 (Springer-Verlag, 1977).Google Scholar
2.Bennett, G., Lectures on matrix transformations of ℓp spaces, in Notes in Banach spaces (Lacey, H. E., ed.) (University of Texas Press, Austin, Texas, 1980).Google Scholar
3.Christensen, J. P. R., Some results with relation to the control measure problem, in Vector space measures and applications II. Lecture Notes in Mathematics No 645 (Springer-Verlag, 1978).Google Scholar
4.Christensen, J. P. R. and Herer, W., On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann. 213 (1975), 203–210.Google Scholar
5.Hunt, R. A., On L (p, q) spaces, Enseignement Math. 12 (1966), 249–274.Google Scholar
6.Johnson, W. B., Maurey, B., Schechtman, G. and Tzafriri, L., Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. No. 217 (Providence, 1979).Google Scholar
7.Kalton, N. J., Topologies on Riesz groups with applications to measure theory, Proc. London Math. Soc. (3) 28 (1974), 253–273.CrossRefGoogle Scholar
8.Kalton, N. J., Linear operators on Lp for 0 < p < l, Trans. Amer. Math. Soc. 259 (1980), 319–355.Google Scholar
9.Kalton, N. J., Isomorphisms between spaces of vector-valued continuous functions, Proc. Edinburgh Math. Soc. 26 (1983), 29–48.CrossRefGoogle Scholar
10.Kalton, N. J., Representations of operators between function spaces, Indiana Univ. Math. J. to appear.Google Scholar
11.Kalton, N. J. and Roberts, J. W., Uniformly exhaustive submeasures and nearly additive set functions, Trans. Amer. Math. Soc. 278 (1983), 803–816.CrossRefGoogle Scholar
12.Krivine, J. L., Théoremes de factorisation dans les espaces reticules, Seminaire Maurey-Schwartz 1973–4, Exposés 22–23, Ecole Polytechnique (Paris).Google Scholar
13.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II, Function spaces (Springer- Verlag, 1979).Google Scholar
14.Maurey, B., Type et cotype dans les espaces munis de structure locales incontitiononelles, Seminaire Maurey-Schwartz 1973–4, Exposés 24–25Ecole Polytechnique (Paris).Google Scholar
15.Maurey, B.Théorèmes de factorisation pour les operateurs linéaires à valeurs dans un espace Lp (Asterisque No 11, 1974).Google Scholar
16.Maurey, B. and Pisier, G., Series de variables aleatoires vectorielles independantes et propriétés geometriques des espaces de Banach, Studia Math. 58 (1976), 45–90.Google Scholar
17.Popa, N., Uniqueness of symmetric structures in Lp (μ) for 0 < p < 1, Rev. Roumaine. Math. Pures. Appl. to appear.Google Scholar
18.Rolewicz, S., Metric linear spaces (PWN, Warsaw 1972).Google Scholar
19.Schaefer, H. H., Banach lattices and positive operators, (Springer-Verlag, 1974).CrossRefGoogle Scholar
20.Talagrand, M., A simple example of a pathological submeasure, Math. Ann. 252 (1980), 97–102.Google Scholar
21.Thomas, G. E. F., On Radon maps with values in arbitrary topological vector spaces and their integral extension (unpublished paper, 1972).Google Scholar