An F-norm on sequences spaces

Mochammad Idris

doi:10.62918/hjma.v3i1.39

Authors

Mochammad Idris Universitas Lambung Mangkurat

DOI:

https://doi.org/10.62918/hjma.v3i1.39

Keywords:

Young Function, F-Norm, Norm, Sequences Spaces

Abstract

On sequences spaces, we can choose a Young function that plays a role in determining their norm structure. In this article, we modify the Young function by replacing its convexity property with concavity to define the F-norm in these spaces. Furthermore, we explore the properties of the modified Young function. Additionally, we investigate the completeness of the space, allowing it to be classified as an F-space (Frechet space).

References

N. M. Abbas, Some Properties On Orlicz Sequence Spaces, Journal of Babylon University/Pure and Applied Sciences 21 (2013) 2340--2345.

Z. W. Birnbaum, W. Orlicz, Uber Die Verallgemeinerung Des Begriffes Der Zueinander Konjugierten Potenzen, Studia Mathematica 3 (1931) 1--67.

Y. Cui, H. Hudzik, R. Kaczmarek, P. Kolwicz, Geometric Properties of F-Normed Orlicz Spaces, Aequat. Math. 93 (2019) 311--343.

S. Ekariani, H. Gunawan, M. Idris, A contractive mapping theorem for the n-normed space of p-summable sequences, J. Math. Analysis (2013) 1--7.

S. Fatimah, A. A. Masta, Ifronika, R. Wafiqoh, Agustine, Generalized Holder’s inequality in Orlicz sequence spaces, In Proceedings of the 7th Mathematics, Science, and Computer Science Education International Seminar (MSCEIS 2019) (2020).

S. Fatimah, C. Kustiawan, E. Sumiaty, A. A. Masta, An inclusion properties of generalized Orlicz sequence spaces, AIP Conf. Proc. 3029, 040019 (2024).

H. Gunawan, The space of $p$-summable sequences and its natural n-norms, Bull. Austral. Math. Soc. 64 (2001) 137--147.

H. Hudzik, D. Pallaschke, On some convexity properties of Orlicz sequence spaces equipped with the Luxemburg norm, Math. Nachr. 186 (1997) 167--185.

M. Idris, S. Ekariani, H. Gunawan, On the space of p-summable sequences, Mat. Vesnik 65 (2013) 58--63.

N. Khusnussaadah, Supama, Completeness of sequence spaces generated by an Orlicz function, Journal of Sciences

and Data Analysis 19 (2019) 1--14.

S. Konca, M. Idris, H. Gunawan, p-summable sequence spaces with inner products, Beu J. Sci. Techn. {5(1) (2015) 37--41.

S. Konca, M. Idris, Equivalence among three 2-norms on the space of p-summable sequences, Journal of Inequalities and Special Functions 7(4) (2016) 218--224.

S. Konca, M. Idris, New 2-norms formed on lp(R) by bounded linear functionals, Euro-Tbilisi-Math.J. 16(4) (2023) 151--165.

M. A. Krasnoselskii, Ja. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff Ltd., 1961.

E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1978.

A. Kufner, O. John, S. Fucik, Function Spaces, Springer 1 edition, 1977.

C. Kustiawan, A. A. Masta, Dasep, E. Sumiaty, S. Fatimah, S. A. Hazmy, Generalized Orlicz sequence spaces, Barekeng:

J. Math. & App. 17(1) (2023) 0427--0438.

L. Maligranda, M. Mastylo, Inclusion Mappings Between Orlicz Sequence Spaces, Journal of Functional Analysis 176 (2000) 264--279.

W. Orlicz, Uber konjugierte Exponentenfolgen, Studia Mathematica 3 (1932) 200--212.

A. Osancliol, A Note On The Definition of An Orlicz Space, Afyon Kocatepe University Journal of Science and Engineering 15 (2015) 1--6.

P. S. Prayoga, S. Fatimah, A. A. Masta, Several Properties of Discrete Orlicz Spaces, Proceedings of the 7th MSCEIS 2019 (2020).

M. M. Rao, Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.

M. M. Rao, Z. D. Ren, Applications Of Orlicz Spaces, CRC Pres 1 edition, 2002.

K. H. Rosen, K. Krithivasan, Discrete mathematics and its applications: with combinatorics and graph theory, Tata McGraw-Hill Education, 2012.

W. Rudin, Functional Analysis 2nd ed., McGraw-Hill, 1991.

E. Savas, R. Savas, Some sequence spaces defined by Orlicz functions, Archivum Mathematicum 40 (2004) 33--40.

N. Sookoo, The F-norm: A generalization of the norm of functional analysis, Journal of Advances in Mathemathics 11(7), (2015) 5388--5396.

N. Sookoo, F-normed spaces and linear operators, Journal of Interdisciplinary Mathematics 24(4) (2021) 911--919.

M. Taqiyuddin, A. A. Masta, Inclusion properties of Orlicz spaces and weak Orlicz spaces generated by concave functions, IOP Conf. Ser.: Mater. Sci. Eng. 288 012103 (2018).

R. Welland, Inclusion relations among Orlicz spaces, Proc. Amer. Math. Soc. 17(1) (1966) 135--139.

X. Zhang, C. Zhang, Weak Orlicz space generated by concave functions, International Conference on Information Science and Technology (ICIST) (2011) 42--44.