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Extensions of the spaces $c$ and ${c}_{0}$ from single to double sequences. (English) Zbl 0781.46025
The author has taken up the study of some sub-spaces of double sequences of complex numbers under both the regular convergence as well as the convergence in Pringsheim’s sense. He shows that the space of regularly convergent sequences is complete under the sup norm and the space of sequences convergent in Psingsheim’s sense is complete under an associated pseudonorm. He has also given some representation theorem for bounded linear functionals on these spaces.
##### MSC:
 46B45 Banach sequence spaces 46A45 Sequence spaces 46B25 Classical Banach spaces in the general theory of normed spaces 40B05 Multiple sequences and series
##### References:
 [1] G. H. Hardy, On the convergence of certain multiple series,Proc. Cambridge Philos. Soc.,19 (1916–1919), 86–95. [2] F. Móricz, Some remarks on the notion of regular convergence of multiple series,Acta. Math. Hungar.,41 (1983), 161–168. · Zbl 0525.40002 · doi:10.1007/BF01994074 [3] W. Rudin,Functional Analysis, McGraw-Hill, Inc. (New York, 1973). [4] K. Yosida,Functional Analysis, Academic Press, Springer (Berlin-Göttingen-Heidelberg, 1965).