---
tag: 799DX4375M
kind: concept
name: Summability field of a matrix method
msc: [40C05, 40H05]
specializes: [XYTN19EPF2]
sources:
  - url: https://en.wikipedia.org/wiki/Silverman%E2%80%93Toeplitz_theorem
---

## 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|M6311DWJ1E]] given by $A$, and it is a [[sequence space|XYTN19EPF2]].
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|71DMQHW6JZ]], which brings functional analysis into summability.

## Explanations

### L1 human

A matrix method takes a sequence, forms weighted averages of its terms, and looks at whether these averages converge.
The summability field is the collection of all sequences for which this works.
For the identity matrix it is just the convergent sequences.
For the Cesàro averages it is bigger: it also contains $1, 0, 1, 0, \dots$, whose averages tend to $1/2$.