Convex combinations of weak*-convergent sequences and the Mackey topology

0.2 MB·

by  Antonio Avilés

CONVEX COMBINATIONS OF WEAK∗-CONVERGENT
SEQUENCES AND THE MACKEY TOPOLOGY
6
1 ANTONIOAVILE´SANDJOSE´ RODR´IGUEZ
0
2
Abstract. A Banach space X is said to have property (K) if every w∗-
n
a convergentsequenceinX∗admitsaconvexblocksubsequencewhichconverges
J withrespecttotheMackeytopology. Westudytheconnectionofthisproperty
1 withstronglyweaklycompactlygeneratedBanachspacesanditsstabilityun-
2 dersubspaces, quotients andℓp-sums. Weextend aresultofFrankiewiczand
Plebanek byprovingthatproperty(K)ispreservedbyℓ1-sumsoflessthanp

] summands. Withoutanycardinalityrestriction,weshowthatproperty(K)is
A stableunderℓp-sumsfor1<p<∞.
F
.
h
t
a 1. Introduction
m
[ A classicalresult by Mazur ensures that if (xn) is a weakly convergentsequence

1 in a Banach space, then there is a convex block subsequence of (xn) which is norm
v convergent. By a convex block subsequence we mean a sequence (yk) of the form
5
y = a x
2 k X n n
8 n∈Ik
5 where (I ) is a sequence of finite subsets of N with max(I )<min(I ) and (a )
k k k+1 n
0
is a sequence of non-negative real numbers such that a = 1 for all k ∈ N.
1. Pn∈Ik n
Mazur’s theorem can be deduced from a general principle asserting that if E is a
0
6 locallyconvexspace(withtopologicaldualE′)andT isanylocallyconvextopology
1 on E which is compatible with the dual pair hE,E′i (meaning that (E,T)′ =E′),
:
v then
i T
X (1.1) C =C for any convex set C ⊆E.
r If X is a Banach space, then (X∗,w∗)′ = X and the strongest locally convex
a
topology on X∗ which is compatible with the dual pair hX∗,Xi is the Mackey
topology µ(X∗,X), i.e. the topology of uniform convergence on weakly compact
subsets of X. When applied to this setting, (1.1) reads as
(1.2) Cw∗ =Cµ(X∗,X) for any convex set C ⊆X∗.
Date:January25,2016.
2000 Mathematics Subject Classification. 46B26,46B50.
Key words and phrases. Mackey topology, convex block subsequence, property (K), strongly
weaklycompactlygenerated space,Grothendieckspace.
Research partially supported by Ministerio de Econom´ıa y Competitividad and FEDER
(project MTM2014-54182-P). This work was also partially supported by the research project
19275/PI/14 funded by Fundaci´on S´eneca - Agencia de Ciencia y Tecnolog´ıa de la Regi´on de
MurciawithintheframeworkofPCTIRM 2011-2014.
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2 ANTONIOAVILE´SANDJOSE´ RODR´IGUEZ
If one assumes that (BX∗,µ(X∗,X)) is metrizable, then (1.2) allows to conclude
that any w∗-convergent sequence in X∗ admits a convex block subsequence which
is µ(X∗,X)-convergent. This is the keystone of our paper:
Definition 1.1. A Banach space X has property (K) if every w∗-convergent se-
quence in X∗ admits a convex block subsequence which is µ(X∗,X)-convergent.
Property (K) was invented by Kwapien´ to provide an alternative approach to
some results of Kalton and Pel czyn´ski [15] on subspaces of L1[0,1]. A weakening
of property (K) was used by Figiel, Johnson and Pel czyn´ski [10] to show that, in
general, the separable complementation property is not inherited by subspaces. It
is not difficult to check that the spaces c and C[0,1] fail property (K). On the
0
other hand, the basic examples of Banach spaces X having property (K) are:
• Schur spaces, i.e. spaces such that weak convergent sequences in X are
norm convergent. In this case, the topologies w∗ and µ(X∗,X) agree on
bounded subsets of X∗.
• Grothendieck spaces, i.e. spaces such that w∗-convergent sequences in X∗
are weakly convergent, like reflexive spaces and ℓ∞. In this case, Mazur’s
theorem ensures that X has property (K).
• Strongly weakly compactly generated (SWCG) spaces, i.e. those for which
there exists a weakly compact set G⊆X such that for every weakly com-
pact set L ⊆ X and every ε > 0 there is n ∈ N such that L ⊆ nG+εB .
X
Thesearepreciselythespacesforwhich(BX∗,µ(X∗,X))ismetrizable,[19].
A typical example is L1(µ) for any finite measure µ.
In this paper we discuss some features of property (K). We point out that this
property is equivalent to reflexivity for Banach spaces not containing subspaces
isomorphic to ℓ1 (Theorem 2.1). We discuss its stability under subspaces and
quotients. The three-space problem for property (K) is considered and we prove
thata BanachspaceX hasproperty(K)wheneverthereis asubspaceY ⊆X such
that Y is Grothendieck and X/Y has property (K) (Theorem 2.6). On the other
hand,we extend the factthat SWCG Banachspaceshaveproperty(K) by proving
that the same holds for Banachspaces which are strongly generatedby less than p
weakly compact sets (Theorem 2.7). Two different proofs of this result are given.
One of them is based on a diagonal argument by Haydon, Levy and Odell [13],
whichis alsoused to show that property (K)is preservedby ℓ1-sums of less than p
summands(Theorem2.14). This is animprovementofa resultby Frankiewiczand
Plebanek [11], who proved the same statement under the additional assumption
that each summand is either L1(µ) for some finite measure µ or a Grothendieck
space. We stress that Pel czyn´ski (unpublished, 1996) proved that the ℓ1-sum of c
copies of L1[0,1]fails property(K), see [10, Example 4.I]. As pointed out in [11], a
similarcounterexamplecanalsobeconstructedfortheℓ1-sumofbcopiesofL1[0,1].
Wefinishthepaperbyprovingthatproperty(K)ispreservedbyarbitraryℓp-sums
for any 1<p<∞ (Theorem 2.15).
CONVEX COMBINATIONS OF WEAK∗-CONVERGENT SEQUENCES 3
Terminology. All our linear spaces are real. Given a sequence (f ) in a linear
n
space, a rational convex block subsequence of (f ) is a sequence (g ) of the form
n k
g = a f
k X n n
n∈Ik
where (I ) is a sequence of finite subsets of N with max(I ) < min(I ) and
k k k+1
(a ) is a sequence of non-negative rational numbers such that a = 1 for
n Pn∈Ik n
all k ∈ N. By a subspace of a Banach space we mean a closed linear subspace.
Given a Banach space Z, its norm is denoted by k·k if needed explicitly. We
Z
write B = {z ∈ Z : kzk ≤ 1}. The dual of Z is denoted by Z∗ and the
Z Z
evaluation of z∗ ∈Z∗ at z ∈Z is denoted by either z∗(z) or hz∗,zi.
Wewillusewithoutexplicitmentionthefollowingelementaryfacts: (i)aBanach
spaceX hasproperty(K)ifandonlyifeveryw∗-convergentsequenceinX∗ admits
a rational convex block subsequence which is µ(X∗,X)-convergent;(ii) in order to
check whether a Banach space X has property (K), it suffices to test on w∗-null
sequences in BX∗ or w∗-null linearly independent sequences.
ThecardinalityofasetS isdenotedby|S|. Thecardinalpisdefinedastheleast
cardinality of a family M of infinite subsets of N having the following properties:
• N is infinite for every finite subfamily N ⊆M.
T
• There is no infinite set A⊆N such that A\M is finite for all M ∈M.
It is known that ω ≤ p ≤ c and that the equality p = c holds under Martin’s
1
Axiom. We refer the reader to [12, 21] for further information on p.
2. Results
The dualball BX∗ ofa BanachspaceX is saidto be w∗-convex block compact if
everysequenceinBX∗ admits aw∗-convergentconvexblocksubsequence. Aresult
ofBourgain[3](cf. [18,Proposition11]and[20])statesthatBX∗ isw∗-convexblock
compactwhenever X contains no subspace isomorphic to ℓ1. As an application we
get the following:
Theorem2.1. LetX beaBanachspacenotcontainingsubspacesisomorphic toℓ1.
Then X is reflexive if and only if it has property (K).
Proof. Itonlyremainstoprovethe“if”part. AssumethatX hasproperty(K).We
willcheckthatBX∗ isweaklycompactbyprovingthatanysequenceinBX∗ admits
a convex block subsequence which is norm convergent (cf. [7, Corollary 2.2]). Let
(x∗n)beasequenceinBX∗. SinceX containsnoisomorphiccopyofℓ1,thereisaw∗-
convergentconvexblocksubsequence(y∗)of(x∗)(seethe paragraphprecedingthe
k n
theorem). By passingto a further convexblock subsequence,not relabeled,we can
assume that (y∗) is µ(X∗,X)-convergent. Then (y∗) is norm convergent, because
k k
the fact that X contains no isomorphic copy of ℓ1 is equivalent to saying that the
normtopologyandµ(X∗,X)agreesequentiallyonX∗ (seee.g. [9,Theorem5.53]).
This shows that BX∗ is weakly compact and so X is reflexive. (cid:3)
In general, property (K) is not inherited by subspaces. For instance, ℓ∞ has
property(K)(becauseitisGrothendieck)andc doesnot(cf. [15,PropositionC]).
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4 ANTONIOAVILE´SANDJOSE´ RODR´IGUEZ
Indeed, let (e∗) be the canonical basis of ℓ1 = c∗. Then (e∗) is w∗-null and does
n 0 n
not admit any µ(ℓ1,c )-null convex block subsequence. For if y∗ = a e∗
0 k Pn∈Ik n n
is any convex block subsequence, then y∗(x ) = 1 for every k ∈ N, where (x ) is
k k k
the weakly null sequence in c defined by x := e (here (e ) denotes the
0 k Pn∈Ik n n
canonical basis of c ).
0
Following [5], we say that a subspace Y of a Banach space X is w∗-extensible
if every w∗-null sequence in Y∗ admits a subsequence which can be extended to a
w∗-null sequence in X∗. Obviously, any complemented subspace is w∗-extensible.
Onthe otherhand,itis notdifficult tocheckthatifX isa Banachspacesuchthat
BX∗ is w∗-sequentially compact (this holds, for instance, if X is weakly compactly
generated, see e.g. [9, Theorem 13.20]), then all subspaces of X are w∗-extensible
(see [22, Theorem 2.1]).
We omit the easy proof of the following result:
Proposition 2.2. Let X be a Banach space and Y ⊆X a w∗-extensible subspace.
If X has property (K), then Y has property (K).
Corollary 2.3. Let X be an SWCG Banach space. Then any subspace of X has
property (K).
Proof. Since X is weaklycompactly generated,any subspace of X is w∗-extensible
(see the comments preceding Proposition2.2). The resultnow followsfromPropo-
sition 2.2 and the fact that any SWCG Banach space has property (K). (cid:3)
In particular, it follows that any subspace of L1(µ) (for a finite measure µ)
has property (K). We stress that there exist non-SWCG subspaces of L1[0,1], like
the one constructed by Mercourakis and Stamati in [17, Section 3] (cf. [2, Theo-
rem 7.5]).
Corollary 2.4. Let X be a Banach space having property (K). Then:
(i) X contains no complemented subspace isomorphic to c .
0
(ii) X contains no subspace isomorphic to c if X has the separable comple-
0
mentation property.
Proof. (i) follows from Proposition2.2 and the failure of property (K) in c . (ii) is
0
a consequence of (i) and Sobczyk’s theorem (see e.g. [9, Theorem 5.11]). (cid:3)
Corollary 2.5. Let L be a compact Hausdorff topological space. Then C(L) is
Grothendieck if and only if it has property (K).
Proof. Combine Corollary 2.4(i) and the fact that C(L) is Grothendieck if and
only if it does not contain complemented subspaces isomorphic to c (see e.g. [6,
0
p. 74]). (cid:3)
We next discuss the behavior of property (K) with respect to quotients.
Theorem 2.6. Let X be a Banach space and Y ⊆X a subspace.
(i) If Y is reflexive and X has property (K), then X/Y has property (K).
(ii) If Y is Grothendieck and X/Y has property (K), then X has property (K).
CONVEX COMBINATIONS OF WEAK∗-CONVERGENT SEQUENCES 5
Proof. Letq :X →X/Y be the quotientmap. Its adjointq∗ :(X/Y)∗ →X∗ is an
isomorphism (and w∗-w∗-homeomorphism) onto Y⊥.
(i) Let (ξ∗) be a w∗-null sequence in (X/Y)∗. Then (q∗(ξ∗)) is w∗-null in X∗.
n n
Since X has property (K), there is a convex block subsequence (η∗) of (ξ∗) such
n n
that (q∗(η∗)) is µ(X∗,X)-null. We claim that (η∗) converges to 0 with respect
n n
to µ((X/Y)∗,X/Y). Indeed, take any weakly compact set L ⊆ X/Y. Since q is
openandLisboundedandweaklyclosed,wecanfindaboundedandweaklyclosed
set L ⊆X such that q(L )=L. Since Y is reflexive and L is weakly compact, L
0 0 0
is weakly compact as well (see e.g. [4, 2.4.b]). Then
lim sup hη∗,zi = lim sup hq∗(η∗),xi =0.
n→∞ z∈L(cid:12)(cid:12) n (cid:12)(cid:12) n→∞ x∈L0(cid:12)(cid:12) n (cid:12)(cid:12)
This shows that (η∗) converges to 0 with respect to µ((X/Y)∗,X/Y).
n
(ii)Let(x∗)beaw∗-nullsequenceinX. Thenthesequenceofrestrictions(x∗| )
n n Y
isw∗-nullinY∗. SinceY isGrothendieck,thereisaconvexblocksubsequence(w∗)
n
of (x∗n) such that kwn∗|YkY∗ →0. Bearing in mind that the map
φ:X∗/Y⊥ →Y∗, φ(x∗+Y⊥):=x∗| ,
Y
is an isomorphism, we can find a sequence (v∗) in Y⊥ such that
n
(2.1) kwn∗ +vn∗kX∗ →0.
Notethat(v∗)isw∗-null(by (2.1)andthefactthat(w∗)isw∗-null). Thenthereis
n n
a w∗-nullsequence (ξ∗) in (X/Y)∗ such thatq∗(ξ∗)=v∗ for all n∈N. Since X/Y
n n n
has property (K), there is a convex block subsequence (η∗) of (ξ∗) converging to 0
n n
with respect to µ((X/Y)∗,X/Y). It is now easy to check that (q∗(η∗)) is a convex
n
blocksubsequenceof(v∗)convergingto0withrespecttoµ(X∗,X). By(2.1),there
n
is a convex block subsequence (w˜∗) of (w∗) (and so of (x∗)) such that
n n n
kw˜n∗ +q∗(ηn∗)kX∗ →0.
Since (q∗(η∗)) is µ(X∗,X)-null, the same holds for (w˜∗). (cid:3)
n n
We saythata family G ofweaklycompactsubsets ofaBanachspace X strongly
generates X if for everyweakly compact set L⊆X and everyε>0 there is G∈G
such that L ⊆ G+εB . We denote by SG(X) the least cardinality of a family
X
of weakly compact subsets of X which strongly generates X. The cardinal SG(X)
coincides with the cofinality of an ordered structure studied in [2]. Note that X is
SWCG if and only if SG(X)=ω (see [19, Theorem 2.1]).
Theorem 2.7. Let X be a Banach space. If SG(X)<p, then X has property (K).
WewillgivetwodifferentproofsofTheorem2.7. Thefirstoneisbasedonduality
arguments inspired by the proof of [19, Theorem 2.1].
Given a topological space T, the character of a point t∈T, denoted by χ(t,T),
is the least cardinality of a neighborhood basis of t.
Lemma 2.8. Let X be a Banach space. Then SG(X)=χ(0,(BX∗,µ(X∗,X))).
6 ANTONIOAVILE´SANDJOSE´ RODR´IGUEZ
Proof. For every weakly compact set L⊆ X and ε> 0, we consider the neighbor-
hood of 0 in (BX∗,µ(X∗,X)) defined by
V(L,ε):={x∗ ∈BX∗ : PL(x∗)<ε},
wherep (x∗):=sup{|x∗(x)|:x∈L}. The collectionofallsets ofthe formV(L,ε)
L
is a basis of neighborhoods of 0 in (BX∗,µ(X∗,X)).
We first prove that SG(X) ≥ χ(0,(BX∗,µ(X∗,X))). Fix a family G of weakly
compact sets strongly generating X such that |G|=SG(X). We claim that
1
V := V G, : G∈G, n∈N
n (cid:16) n(cid:17) o
is a basis of neighborhoods of 0 in (BX∗,µ(X∗,X)). Indeed, take any weakly com-
pact set L ⊆ X and ε > 0. Then there is G ∈ G such that L ⊆ G+ εB , so
2 X
that V(G, 1) ⊆ V(L,ε) whenever 1 ≤ ε. This proves the claim and shows that
n n 2
χ(0,(BX∗,µ(X∗,X)))≤|V|≤|G|=SG(X).
We nowprovethat SG(X)≤χ(0,(BX∗,µ(X∗,X))). LetW be a basisofneigh-
borhoods of 0 in (BX∗,µ(X∗,X)) with |W| = χ(0,(BX∗,µ(X∗,X))). For each
W ∈ W we can choose a weakly compact set L ⊆ X and ε > 0 such that
W W
V(L ,ε )⊆W. BytheKrein-Smulyantheorem(seee.g. [8,p.51,Theorem11]),
W W
we can assume further that each L is absolutely convex. We claim that
W
G :={nL : W ∈W, n∈N}
W
strongly generates X. To this end, fix a weakly compact set L ⊆ X and ε > 0.
There is some W ∈W such that V(L ,ε )⊆V(L,ε). Pick n∈N with n ≥ ε .
W W εW
We will check that
(2.2) L⊆nL +εB .
W X
Bycontradiction,supposethereisx∈Lwhichdoesnotbelongtotheconvexclosed
set nL +εB . By the Hahn-Banach separation theorem, there is x∗ ∈X∗ with
W X
kx∗kX∗ =1 such that
(2.3) x∗(x)>sup{x∗(y): y ∈nL +εB }=np (x∗)+ε.
W X LW
Set c := (εW1 pLW(x∗)+1)−1. Then cx∗ ∈ BX∗ and cx∗ ∈ V(LW,εW) ⊆ V(L,ε).
In particular, cx∗(x)<ε and therefore
ε
x∗(x)< p (x∗)+ε≤np (x∗)+ε,
ε LW LW
W
which contradicts (2.3). Hence (2.2) holds and this completes the proof that G
strongly generates X, as claimed. It follows that
SG(X)≤|G|≤|W|=χ(0,(BX∗,τ(X∗,X)))
and the proof of the lemma is finished. (cid:3)
First proof of Theorem 2.7. Let (x∗n) be a w∗-null sequence in BX∗. Fix k ∈ N.
Define Ak :=coQ{x∗n :n≥k}⊆BX∗ and observe that
w∗ w∗ (1.2) µ(X∗,X) µ(X∗,X)
0∈{x∗ : n≥k} ⊆co{x∗ : n≥k} = co{x∗ : n≥k} =A .
n n n k
CONVEX COMBINATIONS OF WEAK∗-CONVERGENT SEQUENCES 7
Since SG(X) = χ(0,(BX∗,µ(X∗,X))) (Lemma 2.8) is strictly less than p and Ak
is countable, there is a sequence (y∗ ) in A converging to 0 with respect
n,k n∈N k
to µ(X∗,X) (see e.g. [12, 24A]).
Now, using that χ(0,(BX∗,µ(X∗,X)))<p≤b (see e.g. [12, 14B]), we can find
a map ϕ : N → N such that the sequence (y∗ ) converges to 0 with respect
ϕ(k),k
to µ(X∗,X) (cf. [11, Lemma 1]). Finally, notice that y∗ ∈ A for all k ∈ N
ϕ(k),k k
and,therefore,somesubsequenceof(y∗ )isaconvexblocksubsequenceof(x∗).
ϕ(k),k n
This proves that X has property (K). (cid:3)
Our second proof of Theorem 2.7 is based on Lemma 2.9 below, which is taken
from [13, 2A]. Note that the statement given here is slightly different from the
original one (it follows from the proof of [13, 2A]).
Given two sequences (g ) and (h ) in a linear space, we write
n n
(g )(cid:22)(h )
n n
to denote that (g ) is eventually a rational convex block subsequence of (h ), i.e.
n n
there is n ∈N such that (g ) is a rational convex block subsequence of (h ).
0 n n≥n0 n
Lemma 2.9 (Haydon-Levy-Odell). Let κ be a cardinal with κ < p. Let (f ) be a
n
linearly independent sequence in a linear space and, for each α<κ, let (f ) be a
n,α
rational convex block subsequence of (f ). Suppose that for every finite set F ⊆ κ
n
there is a rational convex block subsequence (g ) of (f ) such that (g )(cid:22)(f ) for
n n n n,α
all α∈F. Then there is a rational convex block subsequence (h ) of (f ) such that
n n
(h )(cid:22)(f ) for all α<κ.
n n,α
Lemma2.10. Letκbeacardinalwithκ<p. LetS beasetoflinearlyindependent
sequences in a linear space. For each α<κ, let S ⊆S such that:
α
(i) If (g )∈S and (h )(cid:22)(g ), then (h )∈S .
n α n n n α
(ii) Every sequence of S admits a rational convex block subsequence belonging
to S .
α
Then every sequence of S admits a rational convex block subsequence belonging
to S .
Tα<κ α
Proof. Fix (f )∈S. We will prove the existence of a rational convex block subse-
n
quence of (f ) belonging to S .
n Tα<κ α
Step 1. We will construct by transfinite induction a collection {(f ):α<κ}
n,α
of rational convex block subsequences of (f ) with the following properties:
n
(P ) (f )∈S for all α<κ.
α n,α α
(Q ) (f )(cid:22)(f ) whenever α≤β <κ.
α,β n,β n,α
For α= 0, we just use (ii) to select any rational convex block subsequence of (f )
n
belonging to S . Suppose now that γ < κ and that we have already constructed
0
a collection {(f ) : α < γ} of rational convex block subsequences of (f ) such
n,α n
that (P ) and (Q ) hold whenever α ≤ β < γ. Since |γ| < p, Lemma 2.9
α α,β
ensures the existence of a rational convex block subsequence (f ) of (f ) such
n,γ n
that (f )(cid:22)(f ) for all α<γ. Property (i) yields (f )∈ S ⊆S. Now
n,γ n,α n,γ Tα<γ α
8 ANTONIOAVILE´SANDJOSE´ RODR´IGUEZ
property(ii)impliesthat,bypassingtoafurtherrationalconvexblocksubsequence,
we can assume that (P ) holds. Clearly, (Q ) also holds for every α≤β ≤γ.
γ α,β
Step 2. Since κ < p, another appeal to Lemma 2.9 allows us to extract a
rational convex block subsequence (h ) of (f ) such that (h ) (cid:22) (f ) for every
n n n n,α
α<κ. In particular, (h )∈S for all α<κ (by (i)). (cid:3)
n α
Lemma 2.11. Let X be a Banach space and L⊆X a weakly compact set. If (x∗)
n
is a w∗-convergent sequence in X∗, then there is a convex block subsequenceof (x∗)
n
which converges uniformly on L.
Proof. Wecansupposewithoutlossofgeneralitythat(x∗)isw∗-null. BytheDavis-
n
Figiel-Johnson-Pel czyn´skifactorizationtheorem(seee.g. [9,Theorem13.22]),there
exist a reflexive Banach space Y and a linear continuous map T : Y → X in such
a way that L ⊆ T(B ). Since T∗ : X∗ → Y∗ is w∗-w∗-continuous, (T∗(x∗)) is
Y n
w∗-nullinY∗. SinceY isreflexive,thereisaconvexblocksubsequence(z∗)of(x∗)
n n
such that kT∗(zn∗)kY∗ →0. Bearing in mind that
sup|zn∗(x)|≤ sup |zn∗(T(y))|=kT∗(zn∗)kY∗ for all n∈N,
x∈L y∈BY
we conclude that (z∗) converges to 0 uniformly on L. (cid:3)
n
Remark 2.12. Let X be a Banach space and G a family of weakly compact subsets
ofX whichstronglygeneratesX. Thenaboundedsequence(x∗)inX∗ isµ(X∗,X)-
n
null if and only if it converges to 0 uniformly on any element of G.
Proof. We only have to check the “if” part. To this end, assume without loss of
generality that x∗n ∈ BX∗ for all n ∈N. Take any weakly compact set L⊆ X and
ε>0. Then there is G∈G such that L⊆G+εB . Therefore
X
sup|x∗(x)|≤ sup|x∗(x)|+ε≤2ε
n n
x∈L x∈G
for large enough n. (cid:3)
Second proof of Theorem 2.7. Write κ:=SG(X)<p. Let G be a family of weakly
compactsubsetsofX whichstronglygeneratesX andsuchthat|G|=κ. Enumerate
G = {G : α < κ}. Let S be the set of all w∗-null linearly independent sequences
α
in X∗ and, for each α < κ, let S ⊆ S be the set consisting of all sequences of S
α
whichconvergeuniformlyonG . Conditions(i)and(ii)ofLemma2.10arefulfilled
α
for this choice of S and S (note that (ii) follows from Lemma 2.11).
α
Now, take any (x∗) ∈ S. By Lemma 2.10, (x∗) admits a convex block sub-
n n
sequence (z∗) converging uniformly on each G . Since {G : α < κ} strongly
n α α
generates X, the sequence (z∗) is µ(X∗,X)-null (Remark 2.12). This proves that
n
X has property (K). (cid:3)
To deal with Theorem 2.14 below we need the following fact, which is folklore
and can be deduced, for instance, by an argument similar to that of [8, p. 104,
Theorem 4] (cf. [14, Lemma 7.2]).
CONVEX COMBINATIONS OF WEAK∗-CONVERGENT SEQUENCES 9
Fact 2.13. Let X :=(Li∈IXi)ℓ1 be the ℓ1-sum of a collection {Xi}i∈I of Banach
spaces. For each i ∈ I, let π : X → X be the ith-coordinate projection. Then for
i i
every weakly compact set L⊆X and ε>0 there is a finite set I ⊆I such that
0
kπ (x)k ≤ε for all x∈L.
X i Xi
i∈I\I0
Theorem 2.14. Let {X } be a collection of Banach spaces having property (K).
i i∈I
If |I|<p, then (Li∈IXi)ℓ1 has property (K).
Proof. For notational convenience, we assume that the index set I is a cardinal,
say κ. Write X := (Lα<κXα)ℓ1 and identify X∗ = (Lα<κXα∗)ℓ∞. Denote by
π : X → X and ρ : X∗ → X∗ the αth-coordinate projections for every α < κ.
α α α α
Let G be the family of all weakly compact sets G ⊆ X for which there is a finite
set I ⊆ κ such that π (G) = {0} for every α ∈ κ \ I . From Fact 2.13 it
G α G
follows that G strongly generates X. Therefore, in order to prove that X has
property (K) it suffices to check that any w∗-null linearly independent sequence
inX∗admitsaconvexblocksubsequencewhichconvergesuniformlyonanyelement
of G (Remark 2.12).
LetS bethesetofallw∗-nulllinearlyindependentsequencesinX∗ and,foreach
α<κ, let S ⊆S be the set of all (x∗)∈S such that (ρ (x∗)) is µ(X∗,X )-null.
α n α n α α
Conditions (i) and (ii) of Lemma 2.10 are satisfied for this choice ((i) is immediate
and(ii)holdsbecauseρ isw∗-w∗-continuousandX hasproperty(K)).Therefore,
α α
every (x∗) ∈ S admits a convex block subsequence (y∗) such that (ρ (y∗)) is
n n α n
µ(X∗,X )-null for every α < κ, which clearly implies that (y∗) converges to 0
α α n
uniformly on each element of G. The proof is complete. (cid:3)
On the other hand, property (K) is stable under ℓp-sums whenever 1 <p <∞,
without any restriction on the cardinality of the family:
Theorem 2.15. Let {X } be a collection of Banach spaces having property (K).
i i∈I
Then (Li∈IXi)ℓp has property (K) for any 1<p<∞.
Proof. Write X :=(Li∈IXi)ℓp and identify X∗ =(Li∈IXi∗)ℓq, where 1<q <∞
satisfies 1+1 =1. SinceeveryelementofX∗iscountablysupported,wecanassume
p q
withoutlossofgeneralitythatI iscountable. Let(x∗)beaw∗-nullsequenceinX∗,
n
that is, (x∗) is bounded and for everyi∈I the sequence (ρ (x∗)) is w∗-null in X∗,
n i n i
where ρ : X∗ → X∗ denotes the ith-coordinate projection. Assume without loss
i i
of generality that (x∗) is linearly independent. Since any w∗-null sequence in X∗
n i
admits a rational convex block subsequence which is µ(X∗,X )-null (because X
i i i
has property (K)), an appeal to Lemma 2.10 (in the countable case) allows us to
extractaconvexblocksubsequenceof(x∗),notrelabeled,suchthatforeveryi∈I
n
the sequence (ρ (x∗)) is µ(X∗,X )-null.
i n i i
Define zn :=(kρi(x∗n)kXi∗)i∈I ∈ℓq(I) for all n∈N, so that kznkℓq(I) =kx∗nkX∗.
Since (z ) is a bounded sequence in the reflexive space ℓq(I), it admits a norm
n
convergentconvexblocksubsequence,thatis,thereexistz ∈ℓq(I), asequence(I )
k
offinitesubsetsofNwithmax(I )<min(I )andasequence(a )ofnon-negative
k k+1 n
10 ANTONIOAVILE´SANDJOSE´ RODR´IGUEZ
real numbers with a =1 for all k ∈N, in such a way that
Pn∈Ik n
lim a z −z =0.
k→∞(cid:13)(cid:13) X n n (cid:13)(cid:13)ℓq(I)
(cid:13)n∈Ik (cid:13)
Define a convex block subsequence (y∗) of (x∗) by
k n
y∗ := a x∗ for all k ∈N.
k X n n
n∈Ik
We will check that (y∗) is µ(X∗,X)-null.
k
To this end, fix any weakly compact set L⊆X and ε>0. We can assume that
L⊆B . There is a finite set J ⊆I such that
X
1
(2.4) |ρ˜(z)|q q ≤ε,
(cid:16) X i (cid:17)
i∈I\J
where ρ˜ denotes the ith-coordinate functional on ℓq(I). Choose k ∈ N large
i 0
enough such that
a z −z ≤ε for all k >k .
(cid:13)(cid:13) X n n (cid:13)(cid:13)ℓq(I) 0
(cid:13)n∈Ik (cid:13)
Then (2.4) yields
q 1
q
(2.5) sup ρ˜ a z ≤2ε.
(cid:16) X (cid:12) i(cid:16) X n n(cid:17)(cid:12) (cid:17)
k>k0 i∈I\J(cid:12)(cid:12) n∈Ik (cid:12)(cid:12)
Bearing in mind that
kρi(yk∗)kXi∗ ≤ X ankρi(x∗n)kXi∗ =ρ˜i(cid:16) X anzn(cid:17)
n∈Ik n∈Ik
for every i∈I and k ∈N, from (2.5) we conclude that
1
sup kρ (y∗)kq q ≤2ε.
(cid:16) X i k X∗(cid:17)
k>k0 i∈I\J i
Ho¨lder’s inequality applied to (kρi(yk∗)kXi∗)i∈I ∈ℓq(I) now yields
(2.6) sup X |bi|·kρi(yk∗)kXi∗ ≤2ε for every (bi)i∈I ∈Bℓp(I).
k>k0 i∈I\J
For each i ∈ I, the sequence (ρ (x∗)) is µ(X∗,X )-null and so the same holds
i n i i
for its convex block subsequence (ρ (y∗)). In particular, (ρ (y∗)) converges to 0
i k i k
uniformly on the weakly compact set π (L) ⊆ X , where π : X →X denotes the
i i i i
ith-coordinate projection. Since J is finite, we can find k >k such that
1 0
ε
(2.7) hρ (y∗),π (x)i ≤ for every k >k , i∈J and x∈L.
(cid:12) i k i (cid:12) |J| 1
(cid:12) (cid:12)