# Summability field

*convergence field of a summation method, for a summation method of sequences (cf. Summation methods)*

The set of all sequences summable by the given method. The summability field of any regular matrix summation method (cf. also Regular summation methods) cannot contain all bounded sequences [3]. The summability field of a regular matrix method that sums even one divergent sequence cannot consist of bounded sequences only [4]. However, regular matrix summation methods that sum divergent sequences and whose summability field does not contain bounded divergent sequences do exist. The set of bounded sequences summable by a given method is called the bounded summability field. The summability field of a regular matrix summation method is a complete locally convex metric space with continuous coordinate projections.

#### References

[1] | R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950) |

[2] | G.F. Kangro, "Theory of summability of sequences and series" J. Soviet Math. , 5 : 1 (1970) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 5–70 |

[3] | H. Steinhaus, "Some remarks on the generalization of the concept of limit" Prace Mat. Fiz. , 22 (1911) pp. 121–134 (In Polish) |

[4] | S. Mazur, W. Orlicz, "Sur les méthodes linéaires du sommation" C.R. Acad. Sci. Paris , 196 (1933) pp. 32–34 |

**How to Cite This Entry:**

Summability field. I.I. Volkov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Summability_field&oldid=18353