User:Lethe/sum

Consider the generalization of infinite series to arbitrarily indexed (possibly uncountable) sums. In other words, let a: I → X, where I is any set and X is an abelian topological group. Let F be the set of all finite subsets of I. Note that F is a directed set ordered under inclusion with union as join. We define the sum of the series as the limit

$\sum _{i\in I}a_{i}=\lim _{F}\left\{\sum _{i\in A}a_{i}\,{\bigg |}A\in F\right\}$

if it exists and say that the series a converges. Thus it is the limit of all finite partial sums. There may be uncountably many finite partial sums, so this is not a limit of a sequence of partial sums, but rather of a net. If I is a well-ordered set, for example any ordinal, then one may take the limit of the set of the partial sums of the first n terms. The limit may be defined in this case even when the above definition is not defined. This happens for conditionally convergent series, for example.

If X is a locally convex space, then we may say that a converges absolutely if i→p_α(a_i) converges in R for each α. In this case, the sum does not depend on the order of the sequence. If X is also ordered the sum may be defined simply as

$\sum _{i\in I}a_{i}=\sup _{F}\left\{\sum _{i\in A}a_{i}\,{\bigg |}A\in F\right\}.$

Presumably the Riemann series theorem can be extended to this case, in which case, since the first definition is invariant under the limit of the first definition will not be defined for conditionally convergent series. Thus (I conjecture that) in locally convex spaces, unless the index set is well-ordered and we can make an order dependent definition of the series, the series is convergent iff it is absolutely convergent.

Real sequences

If X = R, then this sum exists only if countably many terms are nonzero. Let

$I_{n}=\left\{i\in I\,{\bigg |}a_{i}>{\frac {1}{n}}\right\}$

be the set of indices whose terms are greater than 1/n. Each I_n is finite. The set of indices whose terms are nonzero is the union of the I_n by the Archimedean principle, and the union of countably many countable sets is countable by the axiom of choice.

This proof goes forward in general first countable topological vector spaces as well, like Banach spaces; define I_n to be those indices whose terms are outside the n-th neighborhood of 0.

One notes with interest that this proof will fail if X does not satisfy the Archimedean property, for example, if it is not first countable. Perhaps one could find convergent uncountable sums in the hyperreals?

Examples

Given a function X→Y, with Y an abelian topological space, then define

$f_{a}(x)={\begin{cases}0,&x\neq a\\f(a)&x=a\\\end{cases}},$

the function whose support is a singleton {a}. Then

$f=\sum _{x\in X}f_{a}$

in the topology of pointwise convergence.