Editing Series (mathematics) (section)

=== Examples ===
* Given a function <math>f : X \to Y</math> into an abelian topological group <math>Y,</math> define for every <math>a \in X,</math> <math display=block>
f_a(x)= \begin{cases}
0 & x\neq a, \\
f(a) & x=a, \\
\end{cases}</math> a function whose [[Support (mathematics)|support]] is a [[Singleton (mathematics)|singleton]] <math>\{a\}.</math> Then <math display=block>f = \sum_{a \in X}f_a</math> in the [[topology of pointwise convergence]] (that is, the sum is taken in the infinite product group <math>\textstyle Y^{X}</math>).
* In the definition of [[partitions of unity]], one constructs sums of functions over arbitrary index set <math>I,</math> <math display=block>
\sum_{i \in I} \varphi_i(x) = 1.
</math> While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given <math>x,</math> only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise.  Actually, one usually assumes more: the family of functions is ''locally finite'', that is, for every <math>x</math> there is a neighborhood of <math>x</math> in which all but a finite number of functions vanish.  Any regularity property of the <math>\varphi_i,</math> such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.
* On the [[first uncountable ordinal]] <math>\omega_1</math> viewed as a topological space in the [[order topology]], the constant function <math>f : \left[0, \omega_1\right) \to \left[0, \omega_1\right]</math> given by <math>f(\alpha) = 1</math> satisfies <math display=block>
\sum_{\alpha \in [0,\omega_1)}\!\!\! f(\alpha) = \omega_1
</math> (in other words, <math>\omega_1</math> copies of 1 is <math>\omega_1</math>) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.