# zbMATH — the first resource for mathematics

Some new sequence spaces defined by a modulus function. (English) Zbl 0946.46007
A function $$f\: [0,\infty)\to [0,\infty ]$$ is called a modulus if (i) $$f(x)=0$$ iff $$x=0$$, (ii) $$f(x+y)\leq f(x)+f(y)$$, (iii) $$f$$ is increasing, (iv) $$f$$ is continuous at $$0$$. The author introduces three linear spaces $$w_0\subset w\subset w_{\infty }$$ of sequences of complex numbers defined by conditions, where absolute value is replaced by a modulus function. He also shows that the spaces $$w_0$$ and $$w$$ are topological with a topology given by a paranorm.
It should be mentioned that there are two other papers by A. Esi (resp. A. ESi and M. Et) with the same title (neither of them contains references to the other two): [J. Inst. Math. Comput. Sci., Math. Ser. 8, No. 2, 81-86 (1995; Zbl 0843.46002), Pure Appl. Math. Sci. 43, No. 1-2, 95-99 (1996; Zbl 0892.46005)]. The results of the paper under review are more general.

##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 40A05 Convergence and divergence of series and sequences 40C05 Matrix methods for summability
##### Keywords:
sequence space; modulus function; paranorm
Full Text:
##### References:
 [1] BANACH S.: Théorie des operations linearies. Chelsea, New York, 1932. [2] LORENTZ G. G.: A contribution to the theory of divergent sequences. Acta Math. 80 (1948), 167-190. · Zbl 0031.29501 [3] RUCKLE W. H.: FK spaces in which the sequence of coordinate vectors in bounded. Canad. J. Math. 25 (1973), 973-978. · Zbl 0267.46008 [4] DAS G.-SAHOO S. K.: On some sequence spaces. J. Math. Anal. Appl. 164 (1992), 381-388. · Zbl 0778.46011 [5] ESI A.: Some new Sequence Spaces Defined by a Modulus and Statistical Convergence. PhD Thesis, Firat University, Graduate School of Natural and Applied Sciences, Department of Mathematics, 1995 · Zbl 0843.46002 [6] MADDOX I. J.: Sequence spaces defined by a modulus. Math. Proc. Cambridge Philos. Soc. 100 (1986), 161-166. · Zbl 0631.46010 [7] MADDOX I. J.: Inclusion between FK spaces and Kuttner’s theorem. Math. Proc. Cambridge Philos. Soc. 101 (1987), 523-527. · Zbl 0631.46009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.