---
tag: Z4H7272S9S
kind: definition
name: Summation of families over arbitrary index sets
msc: [40A05, 46B15]
defines: 70HJTK5VZH
uses: [00A0RPAQ8A, 0BBTH3YSZA, 4DVJQJV0DG]
sources:
  - url: https://en.wikipedia.org/wiki/Series_(mathematics)
---

## Definition

Let $X$ be a [[normed vector space|00A0RPAQ8A]] over $\R$ or $\C$.
Let $I$ be a set, and let $(x_i)_{i \in I}$ be a family of elements of $X$.
For each [[finite subset|0BBTH3YSZA]] $J \subseteq I$ put $s_J = \sum_{i \in J} x_i$.
The family is summable with sum $s \in X$ if the following holds.
For every $\varepsilon > 0$ there is a finite set $J_0 \subseteq I$ with $\norm{s_J - s} < \varepsilon$ for every finite $J$ with $J_0 \subseteq J \subseteq I$.
Then one writes $s = \sum_{i \in I} x_i$.

## Remarks

Equivalently, the [[net|4DVJQJV0DG]] $(s_J)$, indexed by the finite subsets of $I$ ordered by inclusion, converges to $s$.
The sum is unique when it exists.
The same definition works in any Hausdorff abelian topological group, with neighbourhoods of $s$ in place of balls; see [[summation in topological groups|TXS2H001R4]].
If all $x_i$ are nonnegative reals, the family is summable if and only if the finite sums $s_J$ are bounded, and then $s = \sup_J s_J$.
Example: the family $(2^{-m} 3^{-n})_{(m, n) \in \N \times \N}$ is summable with sum $3$.
Its finite sums are bounded by $\sum_{m \le M} 2^{-m} \sum_{n \le M} 3^{-n} \le 2 \cdot \tfrac32$, and the squares $\set{0, \ldots, M}^2$ give sums tending to $3$.
Non-example: the family $((-1)^n / (n+1))_{n \in \N}$ is not summable, although the series $\sum (-1)^n/(n+1)$ converges.
Any finite set $J_0$ can be enlarged by many even indices $n$, which adds more than $1$ to $s_J$, because $\sum 1/(2k+1)$ diverges.
In a Banach space, the family is summable if and only if for every $\varepsilon > 0$ there is a finite $J_0$ with $\norm{s_K} < \varepsilon$ for every finite $K$ disjoint from $J_0$.
A summable family in a normed space has at most countably many nonzero terms.

## Explanations

### L1 human

The sum of a family is the limit of its finite partial sums, where the finite batches of terms grow in every direction at once.
This removes any dependence on the order of the terms.