# zbMATH — the first resource for mathematics

Copies of $$c_{0}$$ and $$\ell_{\infty}$$ in topological Riesz spaces. (English) Zbl 0903.46010
Summary: The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of $$c_{0}$$ or $$\ell_{\infty}$$.

##### MSC:
 46A40 Ordered topological linear spaces, vector lattices 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B42 Banach lattices 28B05 Vector-valued set functions, measures and integrals 40A99 Convergence and divergence of infinite limiting processes
Full Text:
##### References:
 [1] Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics, Vol. 76. · Zbl 0402.46005 [2] Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. · Zbl 0608.47039 [3] G. Buskes and I. Labuda, On Levi-like properties and some of their applications in Riesz space theory, Canad. Math. Bull. 31 (1988), no. 4, 477 – 486. · Zbl 0631.46006 [4] A. Costé, Convergence des séries dans les espaces \?-normés de fonctions mesurables, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 131 – 134 (French, with English and Russian summaries). · Zbl 0233.46037 [5] S. Díaz, On Schwartz’s C-spaces and Orlicz’s O-spaces, Colloq. Math. 64 (1993), 245-251. · Zbl 0821.46007 [6] L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym theorems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 725 – 731 (English, with Russian summary). · Zbl 0243.28011 [7] L. Drewnowski, On subseries convergence in some function spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 797 – 803 (English, with Russian summary). · Zbl 0297.28014 [8] Lech Drewnowski, Un théorème sur les opérateurs de \?_{\infty }(\Gamma ), C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 22, Aii, A967 – A969 (French, with English summary). · Zbl 0323.46014 [9] L. Drewnowski and I. Labuda, The Orlicz-Pettis theorem for topological Riesz spaces, Proc. Amer. Math. Soc. (to appear). · Zbl 0885.40002 [10] -, Spaces of Bochner measurable functions, Preprint (1996). [11] D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974. · Zbl 0273.46035 [12] N. J. Kalton, Exhaustive operators and vector measures, Proc. Edinburgh Math. Soc. (2) 19 (1974/75), 291 – 300. · Zbl 0302.47020 [13] L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. · Zbl 0484.46003 [14] Leonard Y. H. Yap, On a convolution theorem, Acta Sci. Math. (Szeged) 38 (1976), no. 1-2, 203 – 204. · Zbl 0295.43004 [15] Iwo Labuda, Spaces of measurable functions, Comment. Math. Special Issue 2 (1979), 217 – 249. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. · Zbl 0477.46026 [16] Iwo Labuda, Submeasures and locally solid topologies on Riesz spaces, Math. Z. 195 (1987), no. 2, 179 – 196. · Zbl 0601.46006 [17] E. Langford and C. D. Aliprantis, Regularity properties of quotient Riesz seminorms, Nederl. Akad. Wetensch. Proc. Ser. A 78=Indag. Math. 37 (1975), 199 – 212. · Zbl 0306.46013 [18] W. Matuszewska and W. Orlicz, A note on modular spaces. IX, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 801 – 808 (English, with Loose Russian summary). · Zbl 0164.43002 [19] W. Orlicz, Über die Divergenz von allgemeinen Orthogonalreihen, Studia Math. 4 (1933), 27-32. · JFM 59.1012.01 [20] -, On a class of asymptotically divergent sequences of functions, Studia Math. 12 (1951), 286-307. · Zbl 0044.05701 [21] Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13 – 36. · Zbl 0227.46027 [22] Laurent Schwartz, Un théorème de convergence dans les \?^{\?}, 0\le \?<+\infty , C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A704 – A706 (French). · Zbl 0172.40003 [23] A. I. Veksler and V. A. Geĭler, Order completeness and disjoint completeness of linear partially ordered spaces, Sibirsk. Mat. Ž. 13 (1972), 43 – 51 (Russian). · Zbl 0333.06009 [24] Witold Wnuk, Locally solid Riesz spaces not containing \?$$_{0}$$, Bull. Polish Acad. Sci. Math. 36 (1988), no. 1-2, 51 – 56 (English, with Russian summary). · Zbl 0676.46002 [25] Wojbor A. Woyczyński, Sur la convergence des séries dans les espaces de type (\?), C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A1254 – A1257 (French). · Zbl 0175.42402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.