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MAT354H1
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Andresdel Junco
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Lecture

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Mathematics

MAT354H1

Andresdel Junco

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.
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Proposition 2. Suppose a andib are cimplex numbers, i∈Iai=
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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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