Unconditional convergence and weakly compact maps

Díaz, Santiago

doi:10.1090/S0002-9939-1993-1111216-X

Unconditional convergence and weakly compact maps
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Proc. Amer. Math. Soc. 117 (1993), 417-422 Request permission

Abstract:

We characterize the barrelled spaces $E$ such that every linear continuous map from $E$ to ${l_1}$ carries bounded sets into relatively compact sets. These characterizations involve unconditional convergence of series, Edgar’s ordering for Banach spaces, perfect sequence spaces, copies of ${c_0}$, and Orlicz-Pettis topologies.

References

Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
G. A. Edgar, An ordering for the Banach spaces, Pacific J. Math. 108 (1983), no. 1, 83–98. MR 709700, DOI 10.2140/pjm.1983.108.83
Gilles Godefroy, Existence and uniqueness of isometric preduals: a survey, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 131–193. MR 983385, DOI 10.1090/conm/085/983385
Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257, DOI 10.1007/978-3-322-90559-8
N. J. Kalton, Exhaustive operators and vector measures, Proc. Edinburgh Math. Soc. (2) 19 (1974/75), 291–300. MR 388099, DOI 10.1017/S0013091500015571

Topological vector spaces

Gottfried Köthe, Topological vector spaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 237, Springer-Verlag, New York-Berlin, 1979. MR 551623, DOI 10.1007/978-1-4684-9409-9
C. W. McArthur and J. R. Retherford, Some applications of an inequality in locally convex spaces, Trans. Amer. Math. Soc. 137 (1969), 115–123. MR 239391, DOI 10.1090/S0002-9947-1969-0239391-0
I. Tweddle, Unconditional convergence and vector-valued measures, J. London Math. Soc. (2) 2 (1970), 603–610. MR 270113, DOI 10.1112/jlms/2.Part_{4}.603