User:Mgkrupa/Generalizations of Series Proposal

The importance of Series in mathematics has led to many generalizations of notion. Generalizations include asymptotic series, sums assigned to divergent series, series with elements in topological groups (and topological vector spaces in particular), and series with uncountably many terms.

Asymptotic series

Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge, but they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.

Divergent series

Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include Cesàro summation, (C,k) summation, Abel summation, and Borel summation, in increasing order of generality (and hence applicable to increasingly divergent series).

A variety of general results concerning possible summability methods are known. The Silverman–Toeplitz theorem characterizes matrix summability methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns Banach limits.

Summations over arbitrary index sets

Definitions may be given for sums over an arbitrary index set $I$ .^[1] There are two main differences with the usual notion of series: first, there is no specific order given on the set $I$ ; second, this set $I$ may be uncountable. The notion of convergence needs to be strengthened, because the concept of conditional convergence depends on the ordering of the index set.

If $a:I\mapsto G$ is a function from an index set $I$ to a set $G$ , then the "series" associated to $a$ is the formal sum of the elements $a(x)\in G$ over the index elements $x\in I$ denoted by the

\sum _{x\in I}a(x).

When the index set is the natural numbers $I=\mathbb {N}$ , the function $a:\mathbb {N} \mapsto G$ is a sequence denoted by $a(n)=a_{n}$ . A series indexed on the natural numbers is an ordered formal sum and so we rewrite $\sum _{n\in \mathbb {N} }$ as $\sum _{n=0}^{\infty }$ 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

\sum _{n=0}^{\infty }a_{n}=a_{0}+a_{1}+a_{2}+\cdots .

Families of non-negative numbers

When summing a family {a_i}, i ∈ I, of non-negative numbers, one may define

\sum _{i\in I}a_{i}=\sup {\Bigl \{}\sum _{i\in A}a_{i}\,{\big |}A{\text{ finite, }}A\subset I{\Bigr \}}\in [0,+\infty ].

When the supremum is finite, the set of i ∈ I such that a_i > 0 is countable. Indeed, for every n ≥ 1, the set $A_{n}=\{i\in I\,:\,a_{i}>1/n\}$ is finite, because

{\frac {1}{n}}\,{\textrm {card}}(A_{n})\leq \sum _{i\in A_{n}}a_{i}\leq \sum _{i\in I}a_{i}<\infty .

If I is countably infinite and enumerated as I = {i₀, i₁,...} then the above defined sum satisfies

\sum _{i\in I}a_{i}=\sum _{k=0}^{+\infty }a_{i_{k}},

provided the value ∞ is allowed for the sum of the series.

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.

Abelian topological groups

Let a : I → X, where I is any set and X is an abelian Hausdorff topological group. Let F be the collection of all finite subsets of I, with F viewed as a directed set, ordered under inclusion with union as join. Define the sum S of the family a as the limit

S=\sum _{i\in I}a_{i}=\lim {\Bigl \{}\sum _{i\in A}a_{i}\,{\big |}A\in F{\Bigr \}}

if it exists and say that the family a is unconditionally summable. Saying that the sum S is the limit of finite partial sums means that for every neighborhood V of 0 in X, there is a finite subset A₀ of I such that

S-\sum _{i\in A}a_{i}\in V,\quad A\supset A_{0}.

Because F is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.^[2]^[3]

For every W, neighborhood of 0 in X, there is a smaller neighborhood V such that V − V ⊂ W. It follows that the finite partial sums of an unconditionally summable family a_i, i ∈ I, form a Cauchy net, that is, for every W, neighborhood of 0 in X, there is a finite subset A₀ of I such that

\sum _{i\in A_{1}}a_{i}-\sum _{i\in A_{2}}a_{i}\in W,\quad A_{1},A_{2}\supset A_{0}.

When X is complete, a family a is unconditionally summable in X if and only if the finite sums satisfy the latter Cauchy net condition. When X is complete and a_i, i ∈ I, is unconditionally summable in X, then for every subset J ⊂ I, the corresponding subfamily a_j, j ∈ J, is also unconditionally summable in X.

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 X = R.

If a family a in X is unconditionally summable, then for every W, neighborhood of 0 in X, there is a finite subset A₀ of I such that a_i ∈ W for every i not in A₀. If X is first-countable, it follows that the set of i ∈ I such that a_i ≠ 0 is countable. This need not be true in a general abelian topological group (see examples below).

Unconditionally convergent series

Suppose that I = N. If a family a_n, n ∈ N, is unconditionally summable in an abelian Hausdorff topological group X, then the series in the usual sense converges and has the same sum,

\sum _{n=0}^{\infty }a_{n}=\sum _{n\in \mathbf {N} }a_{n}.

By nature, the definition of unconditional summability is insensitive to the order of the summation. When ∑a_n is unconditionally summable, then the series remains convergent after any permutation σ of the set N of indices, with the same sum,

\sum _{n=0}^{\infty }a_{\sigma (n)}=\sum _{n=0}^{\infty }a_{n}.

Conversely, if every permutation of a series ∑a_n converges, then the series is unconditionally convergent. When X is complete, then unconditional convergence is also equivalent to the fact that all subseries are convergent; if X is a Banach space, this is equivalent to say that for every sequence of signs ε_n = ±1, the series

\sum _{n=0}^{\infty }\varepsilon _{n}a_{n}

converges in X.

Series in topological vector spaces

If X is a Topological Vector Space (TVS) and $\left(x_{\alpha }\right)_{\alpha \in A}$ is a (possibly uncountable) family in X then this family is summable^[4] if the limit $\lim _{H\in {\mathcal {F}}(A)}x_{H}$ of the net $\left(x_{H}\right)_{H\in {\mathcal {F}}(A)}$ converges in X, where ${\mathcal {F}}(A)$ is the directed set of all finite subsets of A directed by inclusion $\subseteq$ and $x_{H}:=\sum _{i\in H}x_{i}$ .

It is called absolutely summable if in addition, for every continuous seminorm p on X, the family $\left(p\left(x_{\alpha }\right)\right)_{\alpha \in A}$ is summable. If X is a normable space and if $\left(x_{\alpha }\right)_{\alpha \in A}$ is an absolutely summable family in X, then necessarily all but a countable collection of $x_{\alpha }$ 's are 0. 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 spaces.

Series in Banach and semi-normed spaces

The notion of series can be easily extended to the case of a seminormed space. If x_n is a sequence of elements of a normed space X and if x is in X, then the series Σx_n converges to x in X if the sequence of partial sums of the series $\left(\sum _{n=0}^{N}x_{n}\right)_{N=1}^{\infty }$ converges to x in X; to wit,

\left\|x-\sum _{n=0}^{N}x_{n}\right\|\to 0

as N → ∞.

More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, Σx_n converges to x if the sequence of partial sums converges to x.

If (X, |·|) is a semi-normed space, then the notion of absolute convergence becomes: A series $\sum _{i\in \mathbf {I} }x_{i}$ of vectors in X converges absolutely if

\sum _{i\in \mathbf {I} }\left|x_{i}\right|<+\infty

in which case all but at most countably many of the values $\left|x_{i}\right|$ 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 Dvoretzky & Rogers (1950)).

Well-ordered sums

Conditionally convergent series can be considered if I is a well-ordered set, for example, an ordinal number α₀. One may define by transfinite recursion:

\sum _{\beta <\alpha +1}a_{\beta }=a_{\alpha }+\sum _{\beta <\alpha }a_{\beta }

and for a limit ordinal α,

\sum _{\beta <\alpha }a_{\beta }=\lim _{\gamma \to \alpha }\sum _{\beta <\gamma }a_{\beta }

if this limit exists. If all limits exist up to α₀, then the series converges.

Examples

Given a function f : X→Y, with Y an abelian topological group, define for every a ∈ X
$f_{a}(x)={\begin{cases}0&x\neq a,\\f(a)&x=a,\\\end{cases}}$

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

$f=\sum _{a\in X}f_{a}$
in the topology of pointwise convergence (that is, the sum is taken in the infinite product group Y^X).
In the definition of partitions of unity, one constructs sums of functions over arbitrary index set I,
$\sum _{i\in I}\varphi _{i}(x)=1.$
While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given x, 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 x there is a neighborhood of x in which all but a finite number of functions vanish. Any regularity property of the φ_i, 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 ω₁ viewed as a topological space in the order topology, the constant function f: [0,ω₁) → [0,ω₁] given by f(α) = 1 satisfies
$\sum _{\alpha \in [0,\omega _{1})}f(\alpha )=\omega _{1}$
(in other words, ω₁ copies of 1 is ω₁) only if one takes a limit over all countable partial sums, rather than finite partial sums. This space is not separable.

References

^ Jean Dieudonné, Foundations of mathematical analysis, Academic Press
^ Bourbaki, Nicolas (1998). General Topology: Chapters 1–4. Springer. pp. 261–270. ISBN 978-3-540-64241-1.
^ Choquet, Gustave (1966). Topology. Academic Press. pp. 216–231. ISBN 978-0-12-173450-3.
^ Schaefer 1999, p. 179-180.

Bromwich, T. J. An Introduction to the Theory of Infinite Series MacMillan & Co. 1908, revised 1926, reprinted 1939, 1942, 1949, 1955, 1959, 1965.
Dvoretzky, Aryeh; Rogers, C. Ambrose (1950). "Absolute and unconditional convergence in normed linear spaces". Proc. Natl. Acad. Sci. U.S.A. 36 (3): 192–197. Bibcode:1950PNAS...36..192D. doi:10.1073/pnas.36.3.192. PMC 1063182. PMID 16588972.
Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Boston: Prindle, Weber & Schmidt, ISBN 978-0-87150-341-1
Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).
Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin,New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.
Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.
Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184.
Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Treves, Francois (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.