Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 19 (2024), 57 -- 66

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DIRECTED BASES WITH NET CONVERGENCE

AR. Murugan, C. Ganesa Moorthy and CT. Ramasamy

Abstract. The concept of a basis having a sequence of elements in a topological vector space is extended to a concept of a directed basis having a net of elements in a topological vector space. This article provides a generalization of the classical result which states that every basis in a complete metrizable topological vector space is a Schauder basis.

2020 Mathematics Subject Classification: 46A35; 46A45.
Keywords: Directed set; Net; Schauder basis; Topological vector space.

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References

  1. F. Albiac, and C. Leranoz, Uniqueness of unconditional bases in nonlocally convex \ell1 - products, J. Math. Anal. Appl. 374 (2011), 394 - 401. MR2729229. Zbl 1210.46005.

  2. S. Banach, Theorie des operations lineaires, Monografje Matematyczne, Warszawa, (1932). Zbl 0005.20901.

  3. P. G. Casazza, and N. J. Kalton, Uniqueness of unconditional bases in c0 - products, Studia Math. 133 (5) (1999), 275 - 294. MR1687211. Zbl 0939.46010.

  4. H. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math., 130 (1973), 309 - 317. MR0402468. Zbl 0267.46012.

  5. C. Ganesa Moorthy, and CT. Ramasamy, Pringsheim convergence of double sequences for uniform boundedness principle, Asian - European J. Math. 10 (4) (2017) Article ID : 1750080 (4 pages). MR3718247. Zbl 1386.46004.

  6. B. V. Limaye and M. Zeltser, On the Pringsheim convergence of double series, Proc. Estonian Acad. Sci. 58 (2) (2009), 108 - 121. MR2552922. Zbl 1206.40004.

  7. S. Loganathan and C. Ganesa Moorthy, A net convergence for Schauder double bases, Asian - European J. Math. 9 (1) (2016) Article ID : 1650010 (33 pages). MR3463181. Zbl 1360.46002.

  8. A. Pringsheim, Elementare Theorie der unendliche Doppel-reihen, Sitsungsber Akad. Wiss. Munchen, 27 (1897), 101 - 157. JFM 28.0215.01.

  9. W. Rudin, Functional Analysis, Mc Graw Hill, New York, 1973. MR0365062. Zbl 0253.46001.

  10. I. Singer, Bases in Banach Spaces I, Springer, New York, 1970. MR0298399. Zbl 0198.16601.



AR. Murugan
Department of Mathematics,
Government Arts and Science College (Affiliated to Madurai Kamaraj University)
Vedasandur - 624 702, Dindigul, Tamilnadu, India.
e-mail: armrgn@gmail.com

C. Ganesa Moorthy
Department of Mathematics, Alagappa University
Karaikudi - 630 004, India.
e-mail: ganesamoorthyc@gmail.com

CT. Ramasamy
Department of Mathematics, Alagappa Government Arts College
Karaikudi - 630 003, India.
e-mail: ctrams83@gmail.com

http://www.utgjiu.ro/math/sma