The Classical Sequence Spaces
2.4 Convergence of series
Definition 2.4.1.Let (xn)∞n=1be a sequence in a Banach spaceX. A (formal) series ∞
n=1xn in X is said to beunconditionally convergent if∞
n=1xπ(n) converges for every permutationπofN.
We will see in Chapter 8 that except in finite-dimensional spaces, uncon- ditional convergence is weaker thanabsolute convergence, i.e., convergence of ∞
n=1xn.
Lemma 2.4.2.Given a series ∞
n=1xn in a Banach space X, the following are equivalent:
(a)∞
n=1xn is unconditionally convergent;
2.4 Convergence of series 39 (b) The series ∞
k=1xnk converges for every increasing sequence of integers (nk)∞k=1;
(c) The series∞
n=1nxn converges for every choice of signs(n);
(d) For every > 0 there exists an n so that if F is any finite subset of {n+ 1, n+ 2, . . .} then
j∈F
xj< .
Proof. We will establish only (a)⇒(d) and leave the other easier implications to the reader. Suppose that (d) fails. Then there exists >0 so that for every nwe can find a finite subsetFn of{n+ 1, . . .}with
j∈Fn
xj≥. We will build a permutationπofNso that∞
n=1xπ(n) diverges.
Take n1 = 1 and let A1 = Fn1. Next pick n2 = maxA1 and let B1 = {n1+ 1, . . . , n2} \A1.Now repeat the process takingA2=Fn2,n3= maxA2
andB2={n2+ 1, . . . , n3} \A2.Iterating we generate a sequence (nk)∞k=1and a partition {nk+ 1, . . . , nk+1} =Ak∪Bk. Define π so thatπ permutes the elements of{nk+ 1, . . . , nk+1} in such a way thatAk precedesBk.Then the series∞
n=1xπ(n)is divergent because the Cauchy condition fails.
Definition 2.4.3.A (formal) series∞
n=1xn in a Banach space X isweakly unconditionally Cauchy (WUC) or weakly unconditionally convergent if for every x∗∈X∗ ∞
n=1|x∗(xn)|<∞.
Proposition 2.4.4.Suppose the series∞
n=1xnconverges unconditionally to somexin a Banach space X. Then
(i)∞
n=1xπ(n)=xfor every permutationπ.
(ii)
n∈Axn converges unconditionally for every infinite subsetA ofN. (iii)∞
n=1xn is WUC.
Proof. Parts (i) and (ii) are immediate. For (iii), given x∗ ∈X∗ the scalar series∞
n=1x∗(xπ(n)) converges for every permutationπ. It is a classical the- orem of Riemann that for scalar sequences the series ∞
n=1an converges un- conditionally if and only if it converges absolutely, i.e.,∞
n=1|an|<∞.Thus we have ∞
n=1|x∗(xn)|<∞.
Let us notice that the name “weakly unconditionally convergent” series can be misleading because such series need not be weakly convergent; we will therefore use the term weakly unconditionally Cauchy or more usually its abbreviation (WUC).
Example 2.4.5.The series ∞
n=1en in c0, where (en)∞n=1 is the canonical basis of the space, is WUC but fails to converge weakly (and so it cannot converge unconditionally). In fact, this is in a certain sense the only coun- terexample as we shall see.
In Proposition 2.4.7 we shall prove that WUC series are in a very natural correspondence with bounded operators onc0. Let us first see a lemma.
Lemma 2.4.6.Let ∞
n=1xn be a formal series in a Banach space X. Then the following are equivalent:
(i)∞
n=1xn is WUC.
(ii) There existsC >0 such that for all (ξ(n))∈c00 we have
∞ n=1
ξ(n)xn≤Cmax
n |ξ(n)|. (iii) There existsC >0 such that
n∈F
nxn≤C for any finite subsetF of Nand alln=±1.
Proof. (i)⇒(ii). Put S=
∞
n=1
ξ(n)xn ∈X : ξ= (ξ(n))∈c00, ξ∞≤1
.
The WUC property implies that S is weakly bounded, therefore it is norm- bounded by the Uniform Boundedness principle.
Obviously, (ii) implies (iii). For (iii) ⇒ (i), given x∗ ∈ X∗ let n = sgnx∗(xn). Then for each integerN we have
N n=1
|x∗(xn)|=x∗ N
n=1
nxn≤Cx∗ and therefore the series∞
n=1|x∗(xn)|converges.
Proposition 2.4.7.Let ∞
n=1xn be a series in a Banach space X. Then ∞
n=1xn is WUC if and only if there is a bounded operator T :c0→X with T en=xn.
Proof. If ∞
n=1xn is WUC then the operator T : c00 → X defined by T ξ =∞
n=1ξ(n)xn is bounded for the c0-norm by Lemma 2.4.6. By density T extends to a bounded operatorT :c0→X.
2.4 Convergence of series 41 For the converse, letT :c0 →X be a bounded operator with T en =xn for alln. For eachx∗∈X∗we have
∞ n=1
|x∗(xn)|= ∞ n=1
|x∗(T en)|= ∞ n=1
|T∗(x∗)(en)|, which is finite since∞
n=1en is WUC.
Proposition 2.4.8.Let ∞
n=1xn be a WUC series in a Banach space X. Then ∞
n=1xn converges unconditionally in X if and only if the operator T :c0→X such thatT en=xn is compact.
Proof. Suppose ∞
n=1xn is unconditionally convergent. We will show that limn→∞T −T Sn = 0, where (Sn)∞n=1 are the partial sum projections as- sociated to the canonical basis (en) of c0. Thus, being a uniform limit of finite-rank operators,T will be compact.
Given >0 we use Lemma 2.4.2 to find n=n() so that ifF is a finite subset of {n+ 1, n+ 2, . . .} then
j∈Fxj ≤/2.For everyx∗ ∈X∗ with
x∗ ≤1 we have
{j∈F : x∗(xj)≥0}
x∗(xj)≤ 2,
therefore
j∈F
|x∗(xj)| ≤.
Hence ifξ∈c00withξ∞≤1 it follows that|x∗(T−T Sm)ξ| ≤form≥n and allx∗∈X∗. By density we conclude thatT −T Sm ≤.
Assume, conversely, thatT is compact. Let us consider T∗∗ :c∗∗0 =∞−→X⊂X∗∗.
The restriction of T∗∗ to B∞ is weak∗-to-norm continuous because on a norm compact set the weak∗ topology agrees with the norm topology. Since ∞
n=1eπ(n) converges weak∗ in ∞ for every permutation π, ∞
n=1xn also converges unconditionally inX.
Note that the above argument also implies the following stability prop- erty of unconditionally convergent series with respect to the multiplication by bounded sequences. The proof is left as an exercise.
Proposition 2.4.9.A series ∞
n=1xn in a Banach space X is uncondition- ally convergent if and only if ∞
n=1tnxn converges (unconditionally) for all (tn)∈∞.
The next theorem and its consequences are essentially due to Bessaga and Pelczy´nski in their 1958 paper [12] and represent some of the earliest applications of the basic sequence methods.
Theorem 2.4.10.Suppose T : c0 → X is a bounded operator. Then the following conditions onT are equivalent:
(i)T is compact, (ii)T is weakly compact, (iii)T is strictly singular.
Proof. (i)⇒(ii) is obvious. For (ii)⇒(iii), let us suppose thatT fails to be strictly singular. Then there exists an infinite-dimensional subspace Y of c0 such that T|Y is an isomorphism onto its range. If T is weakly compact this forcesY to be reflexive, contradicting Proposition 2.2.2.
We now consider (iii)⇒(i). Assume thatT fails to be compact. Then, by Proposition 2.4.8,∞
n=1T endoes not converge unconditionally so, by Lemma 2.4.2, there exists > 0 and a sequence of disjoint finite subsets of inte- gers (Fn)∞n=1 so that
k∈FnT ek ≥ for everyn. Let xn =
k∈FnT ek. (xn)∞n=1is weakly null inX since
k∈Fnek is weakly null inc0. Using Propo- sition 1.3.10 we can, by passing to a subsequence of (xn)∞n=1, assume it is basic in X with basis constantK, say. Then forξ= (ξ(n))∞n=1∈c00,
∞ n=1
ξ(n)xn=T ∞
n=1
ξ(n)
k∈Fn
ek≤ Tmax
n∈N|ξ(n)|. On the other hand,
max
n∈N|ξ(n)| ≤2K ∞ n=1
ξ(n)xn.
Thus (xn)∞n=1 is equivalent to the canonical basis of c0 and therefore to (
k∈Fnek)∞n=1.We conclude thatT cannot be strictly singular.
From now on, whenever we say that a Banach space X contains a copy of a Banach spaceY we mean that X contains a closed subspaceE which is isomorphic toY. Using Theorem 2.4.10 we obtain a very nice characterization of spaces that contain a copy ofc0.
Theorem 2.4.11.In order that every WUC series in a Banach space X be unconditionally convergent it is necessary and sufficient that X contains no copy of c0.
Proof. Suppose thatX contains no copy ofc0 and that ∞
n=1xn is a WUC series inX. By Proposition 2.4.7 there exists a bounded operatorT :c0→X such thatT en=xn for alln.T must be strictly singular since every infinite- dimensional subspace of c0 contains a copy of c0 (Proposition 2.2.1) so T is compact by Theorem 2.4.10. Hence the series∞
n=1xnconverges uncondition- ally by Proposition 2.4.8. The converse follows trivially from Example 2.4.5.
2.4 Convergence of series 43 Remark 2.4.12.This theorem of Bessaga and Pelczy´nski is a prototype for exclusion theorems which say that if we can exclude a certain subspace from a Banach space then it will have a particular property. It had considerable influence in suggesting that such theorems might be true. In Chapter 10 we will see a similar and much more difficult result for Banach spaces not containing 1(due to Rosenthal [197]) which when combined with the Bessaga-Pelczy´nski theorem gives a very elegant pair of bookends in Banach space theory. It is also worth noting that the hypothesis that a Banach space fails to containc0
becomes ubiquitous in the theory precisely because of Theorem 2.4.11.
We have seen that a series∞
n=1xnin a Banach spaceXconverges uncon- ditionally in norm if and only if each subseries ∞
k=1xnk does. In particular every subseries of an unconditionally convergent series is weakly convergent.
The Orlicz-Pettis theorem establishes that the converse is true as well. First we see an auxiliary result.
Lemma 2.4.13.Let m0 be the set of all sequences of scalars assuming only finitely many different values. Thenm0 is dense in ∞.
Proof. Leta= (an)∞n=1be a sequence of scalars witha∞≤1. For any >0 pickN∈Nsuch that N1 < . Then the sequenceb= (bn)∞n=1∈m0given by
bn= (sgn an)j
N if j
N ≤ |an| ≤j+ 1
N , j= 1, . . . , N satisfiesa−b∞≤N1 < .
Theorem 2.4.14 (The Orlicz-Pettis Theorem). Suppose ∞
n=1xn is a series in a Banach space X for which every subseries ∞
k=1xnk converges weakly. Then ∞
n=1xn converges unconditionally in norm.
Proof. The hypothesis easily yields that ∞
n=1xn is a WUC series so, by Proposition 2.4.7, there exists a bounded operatorT :c0→XwithT en=xn for alln. We will show thatT is actually compact.
Let us look atT∗∗ :∞ →X∗∗. For everyA⊂Nlet us denote by χA = (χA(k))∞k=1 the element of∞such that χA(k) = 1 ifk∈Aand 0 otherwise.
By hypothesis
n∈Axn converges weakly inX and it follows thatT∗∗(χA)∈ X. The linear span of all such χA consists of the space of scalar sequences taking only finitely many different values, m0, which by Lemma 2.4.13 is dense in ∞. Hence T∗∗ maps ∞ into X. This means that T is a weakly compact operator. Now Theorem 2.4.10 implies thatT is a compact operator and Proposition 2.4.8 completes the proof.
Now, as a corollary, we can give a reciprocal of Proposition 2.4.4 (iii).
Corollary 2.4.15.If a Banach spaceX is weakly sequentially complete then every WUC series inX is unconditionally convergent.
Proof. If ∞
n=1xn is WUC then ∞
n=1x∗(xn) is absolutely convergent for everyx∗∈X∗, which is equivalent to saying that∞
k=1x∗(xnk) converges for each subseries∞
k=1xnkand eachx∗∈X∗. Hence∞
k=1xnkis weakly Cauchy and therefore weakly convergent by hypothesis. We deduce that ∞
n=1xn converges unconditionally in norm by the Orlicz-Pettis theorem.
The Orlicz-Pettis theorem predates basic sequence techniques. It was first proved by Orlicz in 1929 [162] and referenced in Banach’s book [8]. He at- tributes the result to Orlicz in the special case whenX is weakly sequentially complete so that every WUC series has the property of the theorem. However, it seems that Orlicz did know the more general statement. Independently, Pet- tis published a proof in 1938 [178]. Pettis was interested in such a result as a by-product of the study of vector measures. If Σ is a σ-algebra of sets and µ : Σ→ X is a map such that for every x∗ ∈ X∗ the set function x∗◦µ is a (countably additive) measure then the Orlicz-Pettis theorem implies thatµ is countably additive in the norm topology. Thus weakly countably additive set functions are norm countably additive.
This is an attractive theorem and as a result it has been proved, reproved, and generalized many times since then. It is not clear that there is much left to say on this subject! We will suggest some generalizations in the Problems.