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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.24 n.1 Antofagasta mayo 2005

http://dx.doi.org/10.4067/S0716-09172005000100004 

 

Proyecciones
Vol. 24, No 1, pp. 37-48, May 2005.
Universidad Católica del Norte
Antofagasta - Chile

GENERALIZATIONS OF THE ORLICZ-PETTIS THEOREM

CHRISTOPHER STUART
New Mexico State University, USA.

and

CHARLES SWARTZ
New Mexico State University, USA.

Correspondencia a :


ABSTRACT

The Orlicz-Pettis Theorem for locally convex spaces asserts that a series in the space which is subseries convergent in the weak topology is actually subseries convergent in the original topology of the space. A subseries convergent series can be viewed as a multiplier convergent series where the terms of the series are multiplied by elements of the scalar sequence space m0 of sequences with finite range. In this paper we show that the conclusion of the Orlicz-Pettis Theorem holds (and can be strengthened) if the multiplier space m0 is replaced by a sequence space with the signed weak gliding hump property.


References

[1] J. Batt, Applications of the Orlicz-Pettis theorem to operatorvalued measures and compact and weakly compact linear transformations on the space of continuous functions, Revue Roum. Math. Pures Appl., 14, pp. 907-945, (1969).        [ Links ]

[2] G. Bennett, Some inclusion theorems for sequence spaces, Pacific J. of Math., 46, pp. 17-30, (1973).        [ Links ]

[3] G. Bennett and N. Kalton, FK spaces containing c0, Duke Math. J., 39, pp. 561-582, (1972).        [ Links ]

[4] H. Boos, Classical and Modern Methods in Summability, Oxford Univ. Press, Oxford, (2000).        [ Links ]

[5] H. Boos, C. Stuart, and C. Swartz, Gliding hump properties of matrix domains, Analysis Mathematica, 30, pp. 243-257, (2004).        [ Links ]

[6] P. Dierolf, Theorems of Orlicz-Pettis type for locally convex spaces, Man. Math., 20, pp. 73-94, (1977).        [ Links ]

[7] J. Diestel, Sequences and Series in Banach Sapces, Springer-Verlag, N.Y., (1984).        [ Links ]

[8] J. Diestel and J.J. Uhl, Vector Measures, Amer. Math. Soc. Surveys 15, Providence, (1977).        [ Links ]

[9] N. Dinculeanu, Weak compactness and uniform convergence of operators in spaces of Bochner integrable functions, J. Math. Anal.Appl., 109, pp. 372-387, (1985).        [ Links ]

[10] M. Florencio and P. Paul, A note on multiplier convergent series, Casopis Pro Pest. Mat., 113, pp. 421-478, (1988).        [ Links ]

[11] H.G. Garnier, M. DeWilde, and J. Schmets, Analyse Fonctionnelle I, Birkhauser, Basel, (1968).        [ Links ]

[12] N.J. Kalton, The Orlicz-Pettis Theorem, Contemporary Math., Amer. Math. Soc., Providence, (1980).        [ Links ]

[13] N.J. Kalton, Spaces of Compact Operators, Math. Ann., 208, pp. 267-278, (1974).        [ Links ]

[14] P.K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, N.Y., (1981).        [ Links ]

[15] Li Ronglu, Cui Chengri, and Min-Hyung Cho, Invariants on all admissible polar topologies, Chinese Annals of Math., 19, pp. 1-6,(1998).        [ Links ]

[16] C.W. McArthur, On a theorem of Orlicz and Pettis, Pacific J. of Math., 22, pp. 297-302, (1967).        [ Links ]

[17]D. Noll, Sequential completeness and spaces with the gliding humps property, Manuscripta Math., 66, pp. 237-252, (1990).        [ Links ]

[18] W. Orlicz, Beitr¨age zur Theorie der Orthogonalent wichlungen II, Studia Mathematica, 1, pp. 241-255, (1929).        [ Links ]

[19] B.J. Pettis, Integration in Vector Spaces, Trans. of Amer. Math. Soc., 44, pp. 277-304, (1938).        [ Links ]

[20] C. Stuart, Weak Sequential Completeness in Sequence Spaces, Ph.D. Dissertation, New Mexico State University, (1993).        [ Links ]

[21]C. Stuart,Weak sequential completeness of β-duals, Rocky Mountain Journal of Math., Vol. 26, Number 4, Fall, pp. 1559-1568, (1996).        [ Links ]

[22] C. Stuart and C. Swartz, Orlicz-Pettis theorems for multiplier convergent series, Z. Anal. Anwendungen 17, pp. 805-811, (1998).        [ Links ]

[23] C. Swartz, An Introduction to Functional Analysis, Marcel Dekker, N.Y., (1992).        [ Links ]

[24] C. Swartz, Infinite Matrices and the Gliding Hump, World Scientific,(1996).        [ Links ]

[25] C. Swartz, Orlicz-Pettis theorems for multiplier convergent operator valued series, Proyecciones Journal of Mathematics, Volume 23,pp. 61-72, (2004).        [ Links ]

[26] B.L. Thorp, Sequential evaluation convergence, J. London Math. Soc., 44, pp. 201-209, (1969).        [ Links ]

[27] B. Wang, On c-sequences of operators, Studia Scientiarum Math. Hung. 40, pp. 145-150, (2003).        [ Links ]

[28] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, N.Y., (1978).        [ Links ]

[29] Wu Junde and Li Ronglu, An Orlicz-Pettis Theorem with applications in A-spaces, Studia Sci. Math. Hung., 35, pp. 353-358, (1999).        [ Links ]

Received : February 2005. Accepted : April 2005

Christopher Stuart
Department of Mathematical Sciences
New Mexico State University
Las Cruces, New Mexico, 88003
USA

and

Charles Swartz
Department of Mathematical Sciences
New Mexico State University
Las Cruces, New Mexico, 88003
USA