Editing Series (mathematics) (section)

== Summations over general index sets ==

Definitions may be given for infinitary sums over an arbitrary index set <math>I.</math><ref>{{citation|author=Jean Dieudonné|title=Foundations of mathematical analysis|publisher=Academic Press}}</ref>  This generalization introduces two main differences from the usual notion of series: first, there may be no specific order given on the set <math>I</math>; second, the set <math>I</math> may be uncountable. The notions of convergence need to be reconsidered for these, then, because for instance the concept of [[conditional convergence]] depends on the ordering of the index set.

If <math>a : I \mapsto G</math> is a [[Function (mathematics)|function]] from an [[index set]] <math>I</math> to a set <math>G,</math> then the "series" associated to <math>a</math> is the [[formal sum]] of the elements <math>a(x) \in G </math> over the index elements <math>x \in I</math> denoted by the

<math display=block>\sum_{x \in I} a(x).</math>

When the index set is the natural numbers <math>I=\N,</math> the function <math>a : \N \mapsto G</math> is a [[sequence]] denoted by <math>a(n) = a_n.</math> A series indexed on the natural numbers is an ordered formal sum and so we rewrite <math display=inline>\sum_{n \in \N}</math> as <math display=inline>\sum_{n=0}^{\infty}</math> in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers

<math display=block>\sum_{n=0}^{\infty} a_n = a_0 + a_1 + a_2 + \cdots.</math>

=== Families of non-negative numbers ===
When summing a family <math>\left\{a_i : i \in I\right\}</math> of non-negative real numbers over the index set <math>I</math>, define

<math display=block>\sum_{i\in I}a_i = \sup \biggl\{ \sum_{i\in A} a_i\, : A \subseteq I,  A \text{ finite}\biggr\} \in [0, +\infty].</math>

Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the [[counting measure]], which accounts for the many similarities between the two constructions.

When the supremum is finite then the set of <math>i \in I</math> such that <math>a_i > 0</math> is countable.  Indeed, for every <math>n \geq 1,</math> the [[cardinality]] <math>\left|A_n\right|</math> of the set <math>A_n = \left\{i \in I : a_i > 1/n\right\}</math> is finite because

<math display=block>\frac{1}{n} \, \left|A_n\right| = \sum_{i \in A_n} \frac{1}{n} \leq \sum_{i \in A_n} a_i \leq \sum_{i \in I} a_i < \infty.</math>

Hence the set <math>A = \left\{i \in I : a_i > 0\right\} = \bigcup_{n = 1}^\infty A_n</math> is [[Countable set|countable]].

If <math>I</math> is countably infinite and enumerated as <math>I = \left\{i_0, i_1, \ldots\right\}</math> then the above defined sum satisfies

<math display=block>\sum_{i \in I} a_i = \sum_{k=0}^{\infty} a_{i_k},</math>
provided the value <math>\infty</math> is allowed for the sum of the series.

=== Abelian topological groups ===
Let <math>a : I \to X</math> be a map, also denoted by <math>\left(a_i\right)_{i \in I},</math> from some non-empty set <math>I</math> into a [[Hausdorff space|Hausdorff]] [[Abelian group|abelian]] [[topological group]] <math>X.</math> 
Let <math>\operatorname{Finite}(I)</math> be the collection of all [[Finite set|finite]] [[subset]]s of <math>I,</math> with <math>\operatorname{Finite}(I)</math> viewed as a [[directed set]], [[Partially ordered set|ordered]] under [[Inclusion (mathematics)|inclusion]] <math>\,\subseteq\,</math> with [[Union (set theory)|union]] as [[Join (mathematics)|join]]. 
The family <math>\left(a_i\right)_{i \in I},</math> is said to be {{em|[[unconditionally summable]]}} if the following [[Limit of a net|limit]], which is denoted by <math>\textstyle \sum_{i\in I} a_i</math> and is called the {{em|sum}} of <math>\left(a_i\right)_{i \in I},</math> exists in <math>X:</math>

<math display=block>\sum_{i\in I} a_i := \lim_{A \in \operatorname{Finite}(I)} \ \sum_{i\in A} a_i = \lim \biggl\{\sum_{i\in A} a_i \,: A \subseteq I, A \text{ finite }\biggr\}</math>
Saying that the sum <math>\textstyle S := \sum_{i\in I} a_i</math> is the limit of finite partial sums means that for every neighborhood <math>V</math> of the origin in <math>X,</math> there exists a finite subset <math>A_0</math> of <math>I</math> such that

<math display=block>S - \sum_{i \in A} a_i \in V \qquad \text{ for every finite superset} \; A \supseteq A_0.</math>

Because <math>\operatorname{Finite}(I)</math> is not [[Total order|totally ordered]], this is not a [[limit of a sequence]] of partial sums, but rather of a [[Net (mathematics)|net]].<ref name="Bourbaki">{{cite book|title=General Topology: Chapters 1–4|first=Nicolas|last=Bourbaki|author-link=Nicolas Bourbaki|year=1998|publisher=Springer|isbn=978-3-540-64241-1|pages=261–270}}</ref><ref name="Choquet">{{cite book|title=Topology|first=Gustave|last=Choquet|author-link=Gustave Choquet|year=1966|publisher=Academic Press|isbn=978-0-12-173450-3|pages=216–231}}</ref>

For every neighborhood <math>W</math> of the origin in <math>X,</math> there is a smaller neighborhood <math>V</math> such that <math>V - V \subseteq W.</math>  It follows that the finite partial sums of an unconditionally summable family <math>\left(a_i\right)_{i \in I},</math> form a {{em|[[Cauchy net]]}}, that is, for every neighborhood <math>W</math> of the origin in <math>X,</math> there exists a finite subset <math>A_0</math> of <math>I</math> such that

<math display=block>\sum_{i \in A_1} a_i - \sum_{i \in A_2} a_i \in W \qquad \text{ for all finite supersets } \; A_1, A_2 \supseteq A_0,</math>
which implies that <math>a_i \in W</math> for every <math>i \in I \setminus A_0</math> (by taking <math>A_1 := A_0 \cup \{i\}</math> and <math>A_2 := A_0</math>).

When <math>X</math> is [[Complete topological group|complete]], a family <math>\left(a_i\right)_{i \in I}</math> is unconditionally summable in <math>X</math> if and only if the finite sums satisfy the latter Cauchy net condition. When <math>X</math> is complete and <math>\left(a_i\right)_{i \in I},</math> is unconditionally summable in <math>X,</math> then for every subset <math>J \subseteq I,</math> the corresponding subfamily <math>\left(a_j\right)_{j \in J},</math> is also unconditionally summable in <math>X.</math>

When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group <math>X = \R.</math>

If a family <math>\left(a_i\right)_{i \in I}</math> in <math>X</math> is unconditionally summable then for every neighborhood <math>W</math> of the origin in <math>X,</math> there is a finite subset <math>A_0 \subseteq I</math> such that <math>a_i \in W</math> for every index <math>i</math> not in <math>A_0.</math>  If <math>X</math> is a [[first-countable space]] then it follows that the set of <math>i \in I</math> such that <math>a_i \neq 0</math> is countable.  This need not be true in a general abelian topological group (see examples below).

=== Unconditionally convergent series ===
Suppose that <math>I = \N.</math>  If a family <math>a_n, n \in \N,</math> is unconditionally summable in a Hausdorff [[abelian topological group]] <math>X,</math> then the series in the usual sense converges and has the same sum,

<math display=block>\sum_{n=0}^\infty a_n = \sum_{n \in \N} a_n.</math>

By nature, the definition of unconditional summability is insensitive to the order of the summation.  When <math>\textstyle \sum a_n</math> is unconditionally summable, then the series remains convergent after any [[permutation]] <math>\sigma : \N \to \N</math> of the set <math>\N</math> of indices, with the same sum,

<math display=block>\sum_{n=0}^\infty a_{\sigma(n)} = \sum_{n=0}^\infty a_n.</math>

Conversely, if every permutation of a series <math>\textstyle \sum a_n</math> converges, then the series is unconditionally convergent.  When <math>X</math> is [[Complete topological group|complete]] then unconditional convergence is also equivalent to the fact that all subseries are convergent; if <math>X</math> is a [[Banach space]], this is equivalent to say that for every sequence of signs <math>\varepsilon_n = \pm 1</math><!-- this is not about convergence of functions, even less about uniform convergence. -->, the series

<math display=block>\sum_{n=0}^\infty \varepsilon_n a_n</math>

converges in <math>X.</math>

=== Series in topological vector spaces ===
If <math>X</math> is a [[topological vector space]] (TVS) and <math>\left(x_i\right)_{i \in I}</math> is a (possibly [[uncountable]]) family in <math>X</math> then this family is '''summable'''<ref>{{Cite book |last1=Schaefer |first1=Helmut H. |author-link1=Helmut H. Schaefer |title=Topological Vector Spaces |last2=Wolff |first2=Manfred P. |year=1999 |publisher=Springer |isbn=978-1-4612-7155-0 |edition=2nd |series=Graduate Texts in Mathematics |volume=8 |location=New York, NY |pages=179–180}}</ref> if the limit <math>\textstyle \lim_{A \in \operatorname{Finite}(I)} x_A</math> of the [[Net (mathematics)|net]] <math>\left(x_A\right)_{A \in \operatorname{Finite}(I)}</math> exists in <math>X,</math> where <math>\operatorname{Finite}(I)</math> is the [[directed set]] of all finite subsets of <math>I</math> directed by inclusion <math>\,\subseteq\,</math> and <math display=inline>x_A := \sum_{i \in A} x_i.</math>

It is called '''[[absolutely summable]]''' if in addition, for every continuous seminorm <math>p</math> on <math>X,</math> the family <math>\left(p\left(x_i\right)\right)_{i \in I}</math> is summable.
If <math>X</math> is a normable space and if <math>\left(x_i\right)_{i \in I}</math> is an absolutely summable family in <math>X,</math> then necessarily all but a countable collection of <math>x_i</math>’s are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.

Summable families play an important role in the theory of [[nuclear space]]s.

=== Series in Banach and seminormed spaces ===
The notion of series can be easily extended to the case of a [[seminormed space]]. 
If <math>x_n</math> is a sequence of elements of a normed space <math>X</math> and if <math>x \in X</math> then the series <math>\textstyle \sum x_n</math> converges to <math>x</math> in <math>X</math> if the sequence of partial sums of the series <math display=inline>\bigl(\!\!~\sum_{n=0}^N x_n\bigr)_{N=1}^{\infty}</math> converges to <math>x</math> in <math>X</math>; to wit,

<math display=block>\Biggl\|x - \sum_{n=0}^N x_n\Biggr\| \to 0 \quad \text{ as } N \to \infty.</math>

More generally, convergence of series can be defined in any [[Abelian group|abelian]] [[Hausdorff space|Hausdorff]] [[topological group]]. 
Specifically, in this case, <math>\textstyle \sum x_n</math> converges to <math>x</math> if the sequence of partial sums converges to <math>x.</math>

If <math>(X, |\cdot|)</math> is a [[seminormed space]], then the notion of absolute convergence becomes: 
A series <math display=inline>\sum_{i \in I} x_i</math> of vectors in <math>X</math> '''converges absolutely''' if

<math display=block> \sum_{i \in I} \left|x_i\right| < +\infty</math>

in which case all but at most countably many of the values <math>\left|x_i\right|</math> are necessarily zero.

If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of {{harvtxt|Dvoretzky|Rogers|1950}}).

=== Well-ordered sums ===
Conditionally convergent series can be considered if <math>I</math> is a [[well-ordered]] set, for example, an [[ordinal number]] <math>\alpha_0.</math>
In this case, define by [[transfinite recursion]]:

<math display=block>\sum_{\beta < \alpha + 1}\! a_\beta = a_{\alpha} + \sum_{\beta < \alpha} a_\beta</math>

and for a limit ordinal <math>\alpha,</math>

<math display=block>\sum_{\beta < \alpha} a_\beta = \lim_{\gamma\to\alpha}\, \sum_{\beta < \gamma} a_\beta</math>

if this limit exists.  If all limits exist up to <math>\alpha_0,</math> then the series converges.

=== 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.