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Abstract
In this paper, we discuss about completeness property of Orlicz sequence spaces defined by an Orlicz function. Orlicz sequence space is generalization of p-summable sequence space, for every which is also an Orlicz sequence space. Based on the property of convergence sequence on norm space, we define $\Phi$-convergence sequence on Orlicz sequence space. Moreover, we define $\Phi$-Cauchy sequence and $\Phi$-complete on Orlicz sequence space. In this paper, we show the relationship between the (ordinary) convergent sequence, $\Phi$-convergent and $\Phi$-Cauchy sequences. Finally, it will also be shown that Orlicz sequence space is Banach space and $\Phi$-complete space.
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References
- Alexopoulos, J., 2004, A brief introduction to N-functions and Orlicz function spaces, Kent State University, Stark Campus.
- Birnbaum, Z.W. and Orlicz, W., 1931, Über die Verallgemeinerung des Begriffes der zueinander konjugierten Funktionen, Studia Mathematica 3, pp. 1–67.
- Kamthan, P.K. dan Gupta, M., 1981, Sequence Spaces and Series, Marcel Dekker, Inc., New York.
- Khan, V.A., 2008, On A New Sequence Spaces Defined by Orlicz Function, Commun.Fac.Sci.Univ.Ank.Series A1, Vol. 57, No. 2, pp.25-33.
- Kolk, E., 2011, Topologies in Generalized Orlicz Sequence Spaces, Faculty of Science and Mathematics, University of Nis, Serbia, http://www.pmf.ni.ac.rs/filomat.
- Kreyszig, E., 1978, Introductory Functional Analysis with Applications, John Willey & Sons, New York.
- Lindberg, K.J., 1973, On Subspace of Orlicz Sequence Spaces, Studia Math.45:119-146.
- Lindenstrauss, J. dan Tzafriri, L., 1973, On Orlicz Sequence Spaces:III, Israel J.Math.14:368-389.
- Orlicz, W., 1932, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Polon. Sci. A 1932, 8/9, 207-220.
- Rao, M.M. dan Ren, Z.D., 2002, Applications of Orlicz Spaces, Marcel Dekker Inc, New York.
- Savas, E. dan Savas, R., 2004, Some Sequence Spaces Defined by Orlicz Function, Archivum Mathematicum (BRNO), Tomus 40, pp. 33-40.
References
Alexopoulos, J., 2004, A brief introduction to N-functions and Orlicz function spaces, Kent State University, Stark Campus.
Birnbaum, Z.W. and Orlicz, W., 1931, Über die Verallgemeinerung des Begriffes der zueinander konjugierten Funktionen, Studia Mathematica 3, pp. 1–67.
Kamthan, P.K. dan Gupta, M., 1981, Sequence Spaces and Series, Marcel Dekker, Inc., New York.
Khan, V.A., 2008, On A New Sequence Spaces Defined by Orlicz Function, Commun.Fac.Sci.Univ.Ank.Series A1, Vol. 57, No. 2, pp.25-33.
Kolk, E., 2011, Topologies in Generalized Orlicz Sequence Spaces, Faculty of Science and Mathematics, University of Nis, Serbia, http://www.pmf.ni.ac.rs/filomat.
Kreyszig, E., 1978, Introductory Functional Analysis with Applications, John Willey & Sons, New York.
Lindberg, K.J., 1973, On Subspace of Orlicz Sequence Spaces, Studia Math.45:119-146.
Lindenstrauss, J. dan Tzafriri, L., 1973, On Orlicz Sequence Spaces:III, Israel J.Math.14:368-389.
Orlicz, W., 1932, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Polon. Sci. A 1932, 8/9, 207-220.
Rao, M.M. dan Ren, Z.D., 2002, Applications of Orlicz Spaces, Marcel Dekker Inc, New York.
Savas, E. dan Savas, R., 2004, Some Sequence Spaces Defined by Orlicz Function, Archivum Mathematicum (BRNO), Tomus 40, pp. 33-40.