Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
Banach lattice; order continuous norm; embedding of $\ell_1$
Banach lattice; order continuous norm; embedding of $\ell_1$
Summary:
It is known that a Banach lattice with order continuous norm contains a copy of $\ell_1$ if and only if it contains a lattice copy of $\ell_1$. The purpose of this note is to present a more direct proof of this useful fact, which extends a similar theorem due to R.C. James for Banach spaces with unconditional bases, and complements the $c_0$- and $\ell_{\infty}$-cases considered by Lozanovskii, Mekler and Meyer-Nieberg.
References:
[1] Abramovich Y.A., Lozanovskii G.Ya.: On some numerical characterizations of KN-lineals (Russian). Mat. Zametki 14 (1973), 723-732; Transl.: Math. Notes 14 (1973), 973-978. MR 0338727
[2] Abramovich Y.A., Wickstead A.W.: Remarkable classes of unital AM-spaces. J. Math. Anal. Appl. 180 (1993), 398-411. MR 1251867 | Zbl 0792.46004
[3] Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, New York, 1985. MR 0809372 | Zbl 1098.47001
[4] Diaz J.C.: On a three-space problem: noncontainment of $\ell_p$, $1\le p<\infty$, or $c_0$-subspaces. Publ. Mat. (Barcelona) 37 (1993), 127-132. MR 1240928
[5] Diestel J.: A survey of results related to the Dunford-Pettis property. Contemporary Math. 2 (1980), 15-60. MR 0621850 | Zbl 0571.46013
[6] Drewnowski L., Labuda I.: Copies of $c_0$ and $\ell_{\infty}$ in topological Riesz spaces. Trans. Amer. Math. Soc. 350 (1998), 3555-3570. MR 1466947 | Zbl 0903.46010
[7] Kühn B.: Banachverbände mit ordungsstetiger Dualnorm. Math. Z. 167 (1979), 271-277. MR 0539109
[8] Lindenstrauss J.: Weakly compact sets - their topological properties and the Banach spaces they generate. Proc. Symp. Infinite Dim. Topology 1967, Annals Math. Studies, Princeton Univ. Press, 1972. MR 0417761 | Zbl 0232.46019
[9] Lindenstrauss J., Tzafriri L.: Classical Banach Spaces I, Sequence Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 0500056 | Zbl 0362.46013
[10] Lindenstrauss J., Tzafriri L.: Classical Banach Spaces II, Function Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1979. MR 0540367 | Zbl 0403.46022
[11] Lozanovskii G.Ya.: On one result of Shimogaki (in Russian). Theses of Second Conference of the Pedagogical Institutes of Nord-West Region Devoted to Mathematics and Methods of its Teaching, Leningrad, 1970, 43.
[12] Meyer-Nieberg P.: Banach Lattices. Springer-Verlag, Berlin-Heidelberg-New York, 1991. MR 1128093 | Zbl 0743.46015
[13] de Pagter B., Wnuk W.: Some remarks on Banach lattices with non-atomic duals. Indag. Math. (N.S.) 1 (1990), 391-395. MR 1075887 | Zbl 0731.46008
[14] Polyrakis I.: Lattice-subspaces of $C[0,1]$ and positive bases. J. Math. Anal. Appl. 184 (1994), 1-18. MR 1275938 | Zbl 0802.46035
[15] Wnuk W.: Locally solid Riesz spaces not containing $c_0$. Bull. Polish Acad. Sci. Math. 36 (1988), 51-55. MR 0998207
[16] Wnuk W.: Banach Lattices with Order Continuous Norm. Polish Scientific Publishers, Warszawa, 1999.
[17] Wójtowicz M.: The Sobczyk property and copies of $\ell_{\infty}$ in locally convex-solid Riesz spaces. Arch. Math. 75 (2000), 376-379. MR 1785446
Search
Partner of
