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MAT354H1 (1)
Lecture

Absolute summability and unordered infinite sums

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Department
Mathematics
Course
MAT354H1
Professor
Andresdel Junco
Semester
Winter

Description
Absolute summability and unordered infinite sums I will always denote a countable set, finite or infinite. We want to P consider unordered sums of the form z where the z are complex i∈Ii i numbers and I is some abstract countable set. Of course I may be P identified with N and the sum written in the form znbut this n∈N P will not mean the same thing as ∞ zi. The definition of the latter i=1 sum implicitly uses the ordering on N whereas we want to attach a meaning to infinite sums which is independent of any ordering. Now, how will we define unordered sums of complex numbers? Of course we can only define an infinite sum in terms of approximation by finite sums. For convenience when F is a finite subset of I we will P write F = i∈Fzi. P Definition 1. If s ∈ C we will say that i∈Iiconverges to s and P write i∈Ii= s if the following condition is satisfied: ∀ǫ > 0 there is a finite set 0 ⊂ I such that for any finite set F ⊂ I such that F 0 F we have |sF− s| ≤ ǫ. Of course if we replace “ Fs − s| ≤ ǫ” by “ |F − s| < ǫ” we get an equivalent definition which conforms more with the traditions of analysis but we will find it convenient to allow equality because strict inequalities are not preserved in the limit. There is a strong analogy 1 2 with the usual notion of convergence of series if we think of this as saying that the “generalized sequence” of finite partial sums {s } in- F dexed by the partially ordered (by inclusion) set F of finite subsets of I converges to s, in the sense that, given any ǫ > 0, once the index F is large enough (that is, bigger than F )0s iF ǫ-close to s. P We need to check that if i∈Iziconverges then the sum s is unique. ǫ When z,w ∈ C we will frequently use the notation z ∼ w in place of |z − w| ≤ ǫ. P Lemma 1. If z converges to s and to s then s = s . ′ i∈I i Proof: According to the definition, given ǫ > 0 there are finite sets F 0nd G su0h that for all finite F ⊃ F and G ⊃0G we have 0 ǫ ǫ ′ ǫ sF ∼ s and s G ∼ s . Taking H = F ∪ G0it fo0lows that s H ∼ s and ǫ sH ∼ s . Thus |s − s | < 2ǫ and since ǫ > 0 is arbitrary we conclude that s = s . P Lemma 1 permits us to write i∈I zi= s and whenever we use this notation it is to be understood in the sense of definition 1. In fact this is the only possible way to interpret it since there is no order given on I. Whenever {F } in an increasing sequence of finite subsets of I whose union is I we will use the notation F ր I. n 3 P Proposition 1. i∈Izi= s if and only if for all sequences F n I we have s Fn → s as n → ∞. P Proof: One direction is easy: if i∈Izi= s, F n I and F is 0 ǫ as in Definition 1 then F n F ev0ntually, so s Fn ∼ s eventually. (If Pnis a statement depending on n “P holdn eventually” means that P holds for all sufficiently large n, more precisely that there is an N n such that P nolds for all n ≥ N.) For the other direction suppose that for all sequences F ր n we have s F → s as n → ∞. Suppose what we need to prove is false, n namely there is some ǫ > 0 such that for all finite G there is a finite F ⊃ G such that |s − F| > ǫ. Choose any sequence G n ր I and inductively construct a sequence {F }nof finite sets as follows. First choose F ⊃ G such that |s − s| > ǫ. Then choose F ⊃ F ∪ G 1 1 F1 2 1 2 such that |sF2−s| > ǫ, then F 3 F ∪ 2 suc3 that |s F3−s| > ǫ , etc. Continuing in this way we obtain a sequence F ր n (since F ⊃ G n n such that |sFn − s| > ǫ for all n, contradicting our initial assumption. This concludes the proof. P Proposition 2. Suppose a andib are cimplex numbers, i∈Iai= P A, i∈Ibi= B and let c be any complex number. Then the series 4 P P i∈Iai+ biand i∈Ici both converge as well and we have X X a + b = A + B and ca = cA. i i i i∈I i∈I P Moreover we have |A| ≤ i∈Iai|. Proof: All of these statements follow immediately by considering an arbitrary sequencenF ր I. It is also a good exercise to g0ve “ǫ, F proofs” using Definition 1. If z ,i ∈ I are non-negative real numbers we give a different definition i of the unordered sum, which allows an infinite sum, namely X zi= sups F i∈I F where the supremum ranges over all finite subsets of I. The sum always exists but may be ∞. When I = N it is an easy exercise (try it) that X X zn= zn, n∈N n=1 (if we also allow the sum on the right to be infinite) so, in the case of non-negative reals there is no distinction between ordered and un- ordered summation. P We observe that when z ≥ 0 and z = s < ∞, then for each i i∈Ii ǫ > 0 there is a finite F ⊂ I such that for all finite F ⊃ F we have 0 0 ǫ sF ∼ s and also for any finite G disjoint f0om F we have G ≤ s ≤ ǫ. 5 (In fact, both these statements hold once s ∼ s.) These remarks F0 P show that when the ziare non-negative reals and i∈Ii= s < ∞ the two definitions of unordered sums agree. P P We say i∈Iziconverges absolutely if i∈Izi| < ∞. P Theorem 1. i∈Iiconverges if and only if it converges absolutely. Proof: First suppose the sum converges absolutely. Then, fixing any ǫ > 0, there is a finite set0F such that for any finite G disjoint P from F 0e have i∈G|zi| < ǫ. Fix a sequence of finite setn F ր I. Then there an N such that F ⊃ F . It follows that for N ≤ n < m N 0 we have X |s − s | = |s | ≤ |z | < ǫ, Fm Fn Fm\Fn i i∈Fm\Fn since m \Fnis disjoint from 0 . We have shown that Fn} is a Cauchy sequence so sF → s for some s ∈ C. n Now we just need to show that for any other Gnր I we also have sGn → s. Note that Fn∪G n I so there is an M such that FM∪G M ⊃ F . It follows that for n ≥ M we have 0 X X X |sF − sG | = | zi− zi| ≤ |zi| < ǫ, n n i∈Fn\Gn i∈Gn
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