View source for Summability field

''convergence field of a summation method, for a summation method of sequences (cf. [[Summation methods|Summation methods]])''

The set of all sequences summable by the given method. The summability field of any regular [[Matrix summation method|matrix summation method]] (cf. also [[Regular summation methods|Regular summation methods]]) cannot contain all bounded sequences [[#References|[3]]]. The summability field of a regular matrix method that sums even one divergent sequence cannot consist of bounded sequences only [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Steinhaus,   "Some remarks on the generalization of the concept of limit"  ''Prace Mat. Fiz.'' , '''22'''  (1911)  pp. 121–134  (In Polish)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Mazur,   W. Orlicz,   "Sur les méthodes linéaires du sommation"  ''C.R. Acad. Sci. Paris'' , '''196'''  (1933)  pp. 32–34</TD></TR></table>