Summability field of a matrix method

Summary

The summability field of an infinite matrix $A$ is the set of all sequences $x$ that $A$ transforms into a convergent sequence $Ax$. It is the domain of the matrix summability method given by $A$, and it is a sequence space. Many questions of summability theory are questions about summability fields. For instance, one asks whether a summability field contains the space $c$ of convergent sequences or whether two of them are equal. With a natural topology, the summability field is an FK-space, which brings functional analysis into summability.