ON THE EQUIVALENCE OF CONVERGENCE AND 2-NORM CONVERGENCE IN NORMED SPACES

Year 2016, Volume: 29 Issue: 4, 879 - 881, 19.12.2016

Bahri Turan , Fatma Bilici

Abstract

Let (X,∥.∥) be a normed space and ∥.,...,.∥_{G} be the n-norm given by Gähler. In this paper, we show that ∥.∥-convergence and ∥.,.∥_{G}2-convergence are equivalent.

Keywords

Normed space, n-normed space, n-Banach space

References

• Gähler, S., Lineare 2-normierte räume, Math. Nachr. 28, 1-43, (1964).
• Gähler, S., Untersuchungen ¨uber verallgemeinerte m-metrische R¨aume. I, Math. Nachr., 40, 165--189, (1969).
• Gunawan, H., The space of p-summable sequences and its natural n-norms, Bull. Austral. Math. Soc. 64, 137--147, (2001).
• Gunawan, H. and Mashadi, M., On n-normed spaces, Int. J. Math. Math. Sci. 27, 631--639, (2001).
• Raymond, W. and Freese, Yeol Je Cho, Geometry of linear 2-normed spaces, Nova Science publishers, Inc, Newyork, (2001).
• Turan, B. and Bilici, F., On almost 2-normed vector lattice, Preprint.
• White, A., 2-Banach Spaces, Doctoral Dissertation, Saint Louis University, (1968).