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Extensions of the spaces $c$ and $c\sb 0$ 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.

46B45Banach sequence spaces
46A45Sequence spaces
46B25Classical Banach spaces in the general theory of normed spaces
40B05Multiple sequences and series
Full Text: DOI
[1] G. H. Hardy, On the convergence of certain multiple series,Proc. Cambridge Philos. Soc.,19 (1916--1919), 86--95. · Zbl 46.0405.01
[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). · Zbl 0253.46001
[4] K. Yosida,Functional Analysis, Academic Press, Springer (Berlin-Göttingen-Heidelberg, 1965). · Zbl 0126.11504