aa r X i v : . [ m a t h . F A ] F e b ε -WEAKLY PRECOMPACT SETS IN BANACH SPACES JOS´E RODR´IGUEZ
Abstract.
A bounded subset M of a Banach space X is said to be ε -weakly precompact, for a given ε ≥
0, if every sequence ( x n ) n ∈ N in M admits a subsequence ( x n k ) k ∈ N such thatlim sup k →∞ x ∗ ( x n k ) − lim inf k →∞ x ∗ ( x n k ) ≤ ε for all x ∗ ∈ B X ∗ . In this paper we discuss several aspects of ε -weaklyprecompact sets. On the one hand, we give quantitative versions of thefollowing known results: (a) the absolutely convex hull of a weakly pre-compact set is weakly precompact (Stegall), and (b) for any probabilitymeasure µ , the set of all Bochner µ -integrable functions taking valuesin a weakly precompact subset of X is weakly precompact in L ( µ, X )(Bourgain, Maurey, Pisier). On the other hand, we introduce a relativeof a Banach space property considered by Kampoukos and Mercourakiswhen studying subspaces of strongly weakly compactly generated spaces.We say that a Banach space X has property KM w if there is a family { M n,p : n, p ∈ N } of subsets of X such that: (i) M n,p is p -weakly pre-compact for all n, p ∈ N , and (ii) for each weakly precompact set C ⊆ X and for each p ∈ N there is n ∈ N such that C ⊆ M n,p . All subspacesof strongly weakly precompactly generated spaces have property KM w .Among other things, we study the three-space problem and the stabilityunder unconditional sums of property KM w . Introduction
Let X be a real Banach space. A set M ⊆ X is said to be weaklyprecompact (or conditionally weakly compact ) if every sequence in M admitsa weakly Cauchy subsequence. Rosenthal’s ℓ -theorem [39] states that asubset of a Banach space is weakly precompact if (and only if) it is boundedand contains no ℓ -sequence. Recall that a sequence ( x n ) n ∈ N in X is said Mathematics Subject Classification.
Primary: 46B50. Secondary: 46G10.
Key words and phrases. ε -weakly precompact set; ℓ -sequence; strongly weakly pre-compactly generated Banach space; Lebesgue-Bochner space.The research is partially supported by Agencia Estatal de Investigaci´on [MTM2017-86182-P, grant cofunded by ERDF, EU] and
Fundaci´on S´eneca [20797/PI/18]. to be an ℓ -sequence if it is equivalent to the usual basis of ℓ , i.e., it isbounded and there is a constant c > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X n =1 a n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ c m X n =1 | a n | for all m ∈ N and all a , . . . , a m ∈ R . In this case, we will say that ( x n ) n ∈ N isan ℓ -sequence with constant c . Behrends [7] proved a quantitative versionof Rosenthal’s ℓ -theorem which, loosely speaking, says that a bounded setcontaining ℓ -sequences only with small constant must contain a sequencewhich is close to being weakly Cauchy. To state it properly we need tointroduce some terminology.Let ( x n ) n ∈ N be a bounded sequence in X . We denote by clust X ∗∗ ( x n ) theset of all w ∗ -cluster points of ( x n ) n ∈ N in X ∗∗ , and we write δ ( x n ) to denotethe diameter of clust X ∗∗ ( x n ). It is easy to check that δ ( x n ) = sup x ∗ ∈ B X ∗ inf m ∈ N sup n,n ′ ≥ m | x ∗ ( x n ) − x ∗ ( x n ′ ) | = sup x ∗ ∈ B X ∗ lim sup n →∞ x ∗ ( x n ) − lim inf n →∞ x ∗ ( x n ) . Note that δ ( x n ) = 0 if and only if ( x n ) n ∈ N is weakly Cauchy. In a sense, δ ( x n )measures how far ( x n ) n ∈ N is from being weakly Cauchy. This quantitativeapproach to non-weakly Cauchy sequences was considered in [7, 19, 22, 23,24, 33]. An elementary argument shows that if ( x n ) n ∈ N is an ℓ -sequencewith constant c > , then δ ( x n ) ≥ c (see [23, Lemma 5(i)]). Following [7],we say that ( x n ) n ∈ N admits ε - ℓ -blocks , for a given ε >
0, if for every infiniteset N ⊆ N there exist a finite set { n , . . . , n r } ⊆ N and a , . . . , a r ∈ R with P rk =1 | a k | = 1 such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X k =1 a k x n k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε. Clearly, if ( x n ) n ∈ N admits no ℓ -subsequence with constant ε , then ( x n ) n ∈ N admits ε - ℓ -blocks. Conversely, if ( x n ) n ∈ N admits ε - ℓ -blocks, then no subse-quence of it can be an ℓ -sequence with constant c > ε . Behrends’ theorem(see [7, Theorem 3.2]) reads as follows: Theorem 1.1 (Behrends) . Let X be a Banach space, ( x n ) n ∈ N be a boundedsequence in X and ε > . If ( x n ) n ∈ N admits ε - ℓ -blocks, then there is asubsequence ( x n k ) k ∈ N such that δ ( x n k ) ≤ ε . This leads to the following: -WEAKLY PRECOMPACT SETS IN BANACH SPACES 3
Definition 1.2.
Let X be a Banach space and ε ≥ . We say that a set M ⊆ X is ε -weakly precompact if it is bounded and every sequence ( x n ) n ∈ N in M admits a subsequence ( x n k ) k ∈ N such that δ ( x n k ) ≤ ε . Clearly, a set is 0-weakly precompact if and only if it is weakly precom-pact.In this paper we study ε -weakly precompact sets and Banach spaces gen-erated by them in a strong way (see Definition 1.4 below). In order topresent our results we need further background.A Banach space X is said to be strongly weakly precompactly generated ( SWPG for short) if there is a weakly precompact set C ⊆ X with the fol-lowing property: for every weakly precompact set C ⊆ X and for every ε > n ∈ N such that C ⊆ nC + εB X . This class of spaces was introducedin [27] as a natural companion of the class of strongly weakly compactly gen-erated ( SWCG for short) spaces of Schl¨uchtermann and Wheeler [41], whichare those satisfying the same condition but replacing weak precompactnesswith relative weak compactness. Neither of these two classes is closed undersubspaces. Indeed, Mercourakis and Stamati (see [31, § L [0 ,
1] which is not SWCG. The sameexample works for the property of being SWPG, because a Banach spaceis SWCG if and only if it is weakly sequentially complete and SWPG (see[41, Theorem 2.5] and [29, Theorem 2.2]). In an attempt of characterizingsubspaces of SWCG spaces, Kampoukos and Mercourakis [26] studied thefollowing class of Banach spaces (called there spaces having “property (*)”):
Definition 1.3.
We say that a Banach space X has property KM if thereis a family { M n,p : n, p ∈ N } of subsets of X such that: (i) M n,p is p -relatively weakly compact for all n, p ∈ N . (ii) For each relatively weakly compact set C ⊆ X and for each p ∈ N there is n ∈ N such that C ⊆ M n,p . A bounded set M ⊆ X is said to be ε -relatively weakly compact , for agiven ε ≥
0, if M w ∗ ⊆ X + εB X ∗∗ . Property KM stems somehow froma result proved in [16] saying that a Banach space X is a subspace of aweakly compactly generated Banach space if and only if there is a family { M n,p : n, p ∈ N } of subsets of X satisfying condition (i) of Definition 1.3such that X = S n ∈ N M n,p for all p ∈ N . Subspaces of SWCG spaces haveproperty KM (see [26, Proposition 2.15]), but it is unknown whether theconverse holds.In this paper we focus on the following related class of Banach spaces: JOS´E RODR´IGUEZ
Definition 1.4.
We say that a Banach space X has property KM w if thereis a family { M n,p : n, p ∈ N } of subsets of X such that: (i) M n,p is p -weakly precompact for all n, p ∈ N . (ii) For each weakly precompact set C ⊆ X and for each p ∈ N there is n ∈ N such that C ⊆ M n,p . The paper is organized as follows. In Section 2 we discuss some aspectsof ε -weakly precompact sets in arbitrary Banach spaces. On the one hand,we prove that the absolutely convex hull of an ε -weakly precompact setis 2 ε -weakly precompact (Theorem 2.3). This is a quantitative version ofthe well-known result of Stegall (see [40, Addendum]) that the absolutelyconvex hull of a weakly precompact set is weakly precompact. On the otherhand, we investigate ε -weak precompactness in the Lebesgue-Bochner space L ( µ, X ), where X is a Banach space and µ is a probability measure. Weshow that if ( f n ) n ∈ N is a uniformly integrable sequence in L ( µ, X ) suchthat the sequence ( f n ( ω )) n ∈ N is ε -weakly precompact in X for µ -a.e. ω ∈ Ω,then ( f n ) n ∈ N is 2 ε -weakly precompact in L ( µ, X ) (Theorem 2.11). This is aquantitative version of a result due to Bourgain [8], Maurey and Pisier [34].In Section 3 we study Banach spaces having property KM w . This classis closed under subspaces and includes all SWPG spaces (Proposition 3.3)as well as all spaces having property KM (Theorem 3.8). It is shown thata Banach space X has property KM w whenever there is a subspace Y ⊆ X not containing isomorphic copies of ℓ such that X/Y has property KM w (Theorem 3.12). In Subsection 3.3 we discuss the stability of this propertyunder (countable) unconditional sums. While property KM w is preserved by ℓ -sums (Proposition 3.20), in general this is not the case for c -sums or ℓ p -sums when 1 < p < ∞ . Indeed, if E is any Banach space with a normalized1-unconditional basis and separable dual, and ( X m ) m ∈ N is a sequence ofBanach spaces, then the space ( L m ∈ N X m ) E fails property KM w if X m contains isomorphic copies of ℓ for infinitely many m ∈ N (Theorem 3.23).This extends some previous results on property KM and subspaces of SWPGspaces obtained in [26] and [29]. As an application, we show that the Banachspace of Batt and Hiermeyer [6, §
3] fails property KM w (Corollary 3.28).Subsection 3.4 contains some remarks on property KM w within the settingof Lebesgue-Bochner spaces. Finally, in Subsection 3.5 we show that everyBanach space having property KM satisfies the so-called property (K) ofKwapie´n introduced in [25] (see Theorem 3.31). -WEAKLY PRECOMPACT SETS IN BANACH SPACES 5 Notation and terminology.
Given a set S , its cardinality is denoted by | S | and its power set is denoted by P ( S ). All our Banach spaces are real. Byan operator we mean a continuous linear map between Banach spaces. Bya subspace of a Banach space we mean a norm closed linear subspace. Thetopological dual of a Banach space X is denoted by X ∗ and we write w ∗ to denote the weak ∗ -topology on X ∗ . The evaluation of x ∗ ∈ X ∗ at x ∈ X is denoted by either x ∗ ( x ) or h x, x ∗ i . The norm of X is denoted by k · k X or simply k · k . We write B X to denote the closed unit ball of X , i.e., B X = { x ∈ X : k x k ≤ } . Given two sets C , C ⊆ X , its Minkowski sumis C + C := { x + x : x ∈ C , x ∈ C } . Given a set C ⊆ X : • co( C ) (resp., co( C )) denotes the convex (resp., closed convex) hullof C , • aco( C ) (resp., aco( C )) denotes the absolutely convex (resp., closedabsolutely convex) hull of C , • λC := { λx : x ∈ C } for every λ ∈ R , • span( C ) denotes the subspace of X generated by C .Given a sequence ( x n ) n ∈ N in X , by a convex block subsequence of ( x n ) n ∈ N we mean a sequence (˜ x k ) k ∈ N of the form˜ x k = X n ∈ I k a n x n , where the I k ’s are finite subsets of N with max( I k ) < min( I k +1 ) and ( a n ) n ∈ N is a sequence of non-negative real numbers such that P n ∈ I k a n = 1 forall k ∈ N . 2. ε -weakly precompact sets For the sake of easy reference, the following statement gathers two fun-damental facts on ε -weakly precompact sets and ℓ -sequences which werementioned in the introduction. The first one follows from Behrends’ Theo-rem 1.1, while the second one is elementary. Theorem 2.1.
Let X be a Banach space, M ⊆ X be a bounded set and ε > . (i) If M is not is ε -weakly precompact, then it contains an ℓ -sequencewith constant ε . (ii) If M contains an ℓ -sequence with constant ε , then M cannot be ε ′ -weakly precompact for any ≤ ε ′ < ε . JOS´E RODR´IGUEZ
Proof. (i) Let ( x n ) n ∈ N be a sequence in M such that δ ( x n k ) > ε for everysubsequence ( x n k ) k ∈ N . By Theorem 1.1, ( x n ) n ∈ N does not admit ε - ℓ -blocks.Therefore, ( x n ) n ∈ N admits a subsequence which is an ℓ -sequence with con-stant ε .(ii) Let ( x n ) n ∈ N be an ℓ -sequence with constant ε contained in M . Everysubsequence ( x n k ) k ∈ N is also an ℓ -sequence with constant ε and therefore δ ( x n k ) ≥ ε (see [23, Lemma 5(i)]). It follows that M cannot be ε ′ -weaklyprecompact for any 0 ≤ ε ′ < ε . (cid:3) Remark 2.2.
If a Banach space X contains a subspace isomorphic to ℓ ,then B X cannot be ε -weakly precompact for any ≤ ε < .Proof. Fix 0 < η <
1. By James’ ℓ -distortion theorem (see, e.g., [1, The-orem 10.3.1]), there is a normalized sequence in X which is an ℓ -sequencewith constant 1 − η . By Theorem 2.1(ii), B X cannot be ε ′ -weakly precom-pact for any 0 ≤ ε ′ < − η ). (cid:3) Absolutely convex hulls.
It is known that the absolutely convexhull of a weakly precompact subset of a Banach space is weakly precom-pact. This was first proved by Stegall, see [40, Addendum] (cf. [43, Corol-lary 1.1.9] and [38, Corollary B]). The purpose of this subsection is to givea quantitative version of that result, as follows:
Theorem 2.3.
Let X be a Banach space, M ⊆ X and ε ≥ . If M is ε -weakly precompact, then aco( M ) is ε -weakly precompact. Our approach to Theorem 2.3 will follow some ideas of Stegall’s proof ofthe case ε = 0. We first need to introduce further terminology and to provesome auxiliary lemmata. Given a Banach space X and a bounded sequence( x n ) n ∈ N in X , we writeca( x n ) := inf m ∈ N sup n,n ′ ≥ m k x n − x n ′ k , which is a measure of how far ( x n ) n ∈ N is from being norm Cauchy. Thefollowing definition is related in a natural way to the classical measures ofnon-compactness of Hausdorff and Kuratowski (see, e.g., [22, § Definition 2.4.
Let X be a Banach space, M ⊆ X and ε ≥ . We saythat M is ε -precompact if it is bounded and every sequence ( x n ) n ∈ N in M admits a subsequence ( x n k ) k ∈ N such that ca( x n k ) ≤ ε . -WEAKLY PRECOMPACT SETS IN BANACH SPACES 7 Lemma 2.5.
Let X be a Banach space, M ⊆ X and ε ≥ . Then M is ε -precompact (resp., ε -weakly precompact) if and only if it is ε ′ -precompact(resp., ε ′ -weakly precompact) for every ε ′ > ε .Proof. The “only if” part is obvious and the “if” part follows from a stan-dard diagonalization argument. (cid:3)
The following lemma is a quantitative version of Mazur’s classical resultthat the absolutely convex hull of a relatively norm compact subset of aBanach space is relatively norm compact (see, e.g., [14, p. 51, Theorem 12]).
Lemma 2.6.
Let X be a Banach space, M ⊆ X and ε ≥ . If M is ε -precompact, then aco( M ) is ε -precompact.Proof. Fix ε ′ > ε and take η > ε + 2 η ≤ ε ′ .Observe that there is a finite set F ⊆ X such thatsup x ∈ M min x ′ ∈ F k x − x ′ k ≤ ε + η. Indeed, otherwise we could construct by induction a sequence ( x n ) n ∈ N in M such that k x n − x m k > ε + η whenever n = m , and therefore ca( x n k ) ≥ ε + η for every subsequence ( x n k ) k ∈ N , which contradicts that M is ε -precompact.Then K := aco( F ) is norm compact and(2.1) sup x ∈ aco( M ) min x ′ ∈ K k x − x ′ k ≤ ε + η. Now let ( y n ) n ∈ N be a sequence in aco( M ). By (2.1), there is a sequence( z n ) n ∈ N in K such that k y n − z n k ≤ ε + η for all n ∈ N . Since K is normcompact, ( z n ) n ∈ N admits a subsequence ( z n k ) k ∈ N such that k z n k − z n k ′ k ≤ η and so k y n k − y n k ′ k ≤ ε + 2 η ≤ ε ′ for all k, k ′ ∈ N . Hence ca( y n k ) ≤ ε ′ .This shows that aco( M ) is ε ′ -precompact. As ε ′ > ε is arbitrary, aco( M ) is ε -precompact (Lemma 2.5). (cid:3) We will also need the following simple lemma.
Lemma 2.7.
Let T : X → Y be an operator between the Banach spaces X and Y , M ⊆ X and ε ≥ . If M is ε -precompact (resp., ε -weakly precom-pact), then T ( M ) is k T k ε -precompact (resp., k T k ε -weakly precompact).Proof. Just bear in mind that, for any bounded sequence ( x n ) n ∈ N in X , wehave ca( T ( x n )) ≤ k T k ca( x n ) and δ ( T ( x n )) ≤ k T k δ ( x n ). (cid:3) Given a compact Hausdorff topological space K and a regular Borel prob-ability measure µ on K , the operator i µ : C ( K ) → L ( µ ) that sends each JOS´E RODR´IGUEZ function to its equivalence class is completely continuous (i.e., it mapsweakly Cauchy sequences to norm convergent sequences), as an immedi-ate consequence of Lebesgue’s dominated convergence theorem. We nextprovide a quantitative version of this fact.
Lemma 2.8.
Let K be a compact Hausdorff topological space, µ be a regularBorel probability measure on K , and ε ≥ . If M ⊆ C ( K ) is ε -weaklyprecompact, then i µ ( M ) is ε -precompact in L ( µ ) .Proof. By Lemma 2.5, it suffices to check that i µ ( M ) is ε ′ -precompact forevery ε ′ > ε . Write α := sup {k f k C ( K ) : f ∈ M } and choose η > αη + ε ≤ ε ′ . Let ( f n ) n ∈ N be a sequence in M . Since M is ε -weakly precompact, by passing to a subsequence we can assume that δ ( f n ) ≤ ε . By [22, Proposition 9.1], there is a closed set L ⊆ K such that µ ( K \ L ) ≤ η and the sequence of restrictions ( f n | L ) n ∈ N in C ( L ) satisfiesca( f n | L ) ≤ δ ( f n ), hence ca( f n | L ) ≤ ε . Observe that for each k, k ′ ∈ N wehave k i µ ( f k ) − i µ ( f k ′ ) k L ( µ ) = Z K | f k − f k ′ | dµ = Z K \ L | f k − f k ′ | dµ + Z L | f k − f k ′ | dµ ≤ αη + (cid:13)(cid:13) f k | L − f k ′ | L (cid:13)(cid:13) C ( L ) , hence ca( i µ ( f n )) ≤ αη + ca( f n | L ) ≤ αη + ε ≤ ε ′ . This shows that i µ ( M ) is ε ′ -precompact for every ε ′ > ε . (cid:3) Lemma 2.9.
Let (Ω , Σ , µ ) be a probability space, j µ : L ∞ ( µ ) → L ( µ ) bethe inclusion operator, and ε ≥ . If M ⊆ L ∞ ( µ ) is ε -weakly precompact,then j µ ( M ) is ε -precompact in L ( µ ) .Proof. Let K be the Stone space of the measure algebra of µ and let ˜ µ bethe regular Borel probability on K induced by µ . Then there exist isometricisomorphisms I ∞ : L ∞ ( µ ) → C ( K ) and I : L (˜ µ ) → L ( µ ) such that thediagram L ∞ ( µ ) j µ / / I ∞ (cid:15) (cid:15) L ( µ ) C ( K ) i ˜ µ / / L (˜ µ ) I O O -WEAKLY PRECOMPACT SETS IN BANACH SPACES 9 commutes. Since M is ε -weakly precompact and k I ∞ k = 1, the set I ∞ ( M )is ε -weakly precompact (Lemma 2.7). Now, we can apply Lemma 2.8 todeduce that i ˜ µ ( I ∞ ( M )) is ε -precompact. Since k I k = 1, we conclude that j µ ( M ) is ε -precompact (Lemma 2.7 again). (cid:3) We are now ready to prove Theorem 2.3.
Proof of Theorem 2.3.
By Lemma 2.5, it suffices to check that aco( M ) is ε ′ -weakly precompact for every ε ′ > ε . To this end we will apply The-orem 2.1(i), that is, we will show that if ( x n ) n ∈ N is an ℓ -sequence withconstant C > M ), then C < ε ′ .Let ( r n ) n ∈ N be the sequence of Rademacher functions on [0 ,
1] and let T : span( { x n : n ∈ N } ) → L ∞ [0 , T ( x n ) = r n for all n ∈ N , so that k T k ≤ C . Since L ∞ [0 ,
1] is isometrically injective (see, e.g., [1, Propo-sition 4.3.8(ii)]), T extends to an operator ˜ T : X → L ∞ [0 ,
1] such that k ˜ T k = k T k . Define T := j ◦ ˜ T : X → L [0 , , where j : L ∞ [0 , → L [0 ,
1] is the inclusion operator.On the one hand, r n = T ( x n ) ∈ T (aco( M )) = aco( T ( M )) for all n ∈ N and we have k r n − r n ′ k L [0 , = 1 whenever n = n ′ . Therefore, aco( T ( M ))cannot be ε ′′ -precompact for any 0 ≤ ε ′′ < M is ε -weakly precompact and k ˜ T k ≤ C , the set˜ T ( M ) is εC -weakly precompact in L ∞ [0 ,
1] (Lemma 2.7). Now, Lemma 2.9ensures that T ( M ) is εC -precompact in L [0 ,
1] and therefore aco( T ( M )) is εC -precompact as well (Lemma 2.6). It follows that 1 ≤ εC and so C < ε ′ ,as we wanted. (cid:3) Question 2.10.
Is constant optimal in Theorem 2.3? Lebesgue-Bochner spaces.
Let (Ω , Σ , µ ) be a probability space and X be a Banach space. The characteristic function of any A ∈ Σ is denotedby χ A . Given a function f : Ω → X , we denote by k f ( · ) k X the real-valued function on Ω defined by ω
7→ k f ( ω ) k X . As usual, L ∞ ( µ, X ) is theBanach space of all (equivalence classes of) strongly µ -measurable functions f : Ω → X which are µ -essentially bounded, equipped with the norm k f k L ∞ ( µ,X ) := (cid:13)(cid:13) k f ( · ) k X (cid:13)(cid:13) L ∞ ( µ ) . We denote by L ( µ, X ) the Banach space of all (equivalence classes of)Bochner µ -integrable functions f : Ω → X , equipped with the norm k f k L ( µ,X ) := Z Ω k f ( · ) k X dµ. A set W ⊆ L ( µ, X ) is said to be uniformly integrable if it is bounded andfor every ε > δ > f ∈ W R A k f ( · ) k X dµ ≤ ε for every A ∈ Σ with µ ( A ) ≤ δ . The simplest example of a uniformly integrable setis L ( M ) := { f ∈ L ( µ, X ) : f ( ω ) ∈ M for µ -a.e. ω ∈ Ω } for a bounded set M ⊆ X . Every weakly precompact subset of L ( µ, X ) isuniformly integrable (see, e.g., [14, p. 104, Theorem 4]), but the conversedoes not hold in general. The most penetrating study of weak precom-pactness in Lebesgue-Bochner spaces was made by Talagrand [44]. Herewe will focus on an earlier result proved independently by Bourgain (see [8,Theorem 8]), Maurey and Pisier [34]: if ( f n ) n ∈ N is a uniformly integrable se-quence in L ( µ, X ) such that the sequence ( f n ( ω )) n ∈ N is weakly precompactin X for µ -a.e. ω ∈ Ω , then ( f n ) n ∈ N is weakly precompact in L ( µ, X ) (cf. [44, Corollary 10] ). The purpose of this subsection is to prove the followingquantitative version of the Bourgain-Maurey-Pisier theorem:
Theorem 2.11.
Let (Ω , Σ , µ ) be a probability space, X be a Banach space,and ε ≥ . Let F : Ω → P ( X ) be a multi-function such that F ( ω ) is ε -weakly precompact for µ -a.e. ω ∈ Ω . Write S ( F ) := { f ∈ L ( µ, X ) : f ( ω ) ∈ F ( ω ) for µ -a.e. ω ∈ Ω } to denote the set of all (equivalence classes of ) Bochner µ -integrable selectorsof F . Then: (i) If ( f n ) n ∈ N is a uniformly integrable sequence in S ( F ) , then ( f n ) n ∈ N cannot be an ℓ -sequence with constant C > ε . (ii) Every uniformly integrable subset of S ( F ) is ε -weakly precompactin L ( µ, X ) . When applied to constant multi-functions, the previous theorem yields:
Corollary 2.12.
Let (Ω , Σ , µ ) be a probability space, X be a Banach space,and ε ≥ . If M ⊆ X is ε -weakly precompact, then L ( M ) is ε -weaklyprecompact in L ( µ, X ) . To prove Theorem 2.11 we will follow the approach to the Bourgain-Maurey-Pisier theorem which can be found in [10, § -WEAKLY PRECOMPACT SETS IN BANACH SPACES 11 previous lemmata. The first one is a straightforward application of Fatou’slemma. Lemma 2.13.
Let (Ω , Σ , µ ) be a probability space and ( g n ) n ∈ N be a boundedsequence in L ∞ ( µ ) . Then Z Ω lim inf n →∞ g n dµ ≤ lim inf n →∞ Z Ω g n dµ ≤ lim sup n →∞ Z Ω g n dµ ≤ Z Ω lim sup n →∞ g n dµ. Proof.
Write C := sup n ∈ N k g n k L ∞ ( µ ) < ∞ . Now, we can apply Fatou’slemma to the sequences ( C + g n ) n ∈ N and ( C − g n ) n ∈ N to get the desiredinequalities. (cid:3) Lemma 2.14.
Let (Ω , Σ , µ ) be a probability space, ( h n ) n ∈ N be a boundedsequence in L ∞ ( µ ) , and ε ≥ . If lim sup n →∞ h n − lim inf n →∞ h n ≤ ε µ -a.e.,then lim sup m →∞ Z Ω (cid:12)(cid:12)(cid:12) h m − lim inf n →∞ h n (cid:12)(cid:12)(cid:12) dµ ≤ ε. Proof.
Write g m := (cid:12)(cid:12)(cid:12) h m − lim inf n →∞ h n (cid:12)(cid:12)(cid:12) for all m ∈ N .Then ( g m ) m ∈ N is a bounded sequence in L ∞ ( µ ) with lim inf m →∞ g m = 0.We havelim sup m →∞ g m = lim sup m →∞ g m − lim inf m →∞ g m = inf n ∈ N sup m,m ′ ≥ n | g m − g m ′ |≤ inf n ∈ N sup m,m ′ ≥ n | h m − h m ′ | = lim sup m →∞ h m − lim inf m →∞ h m ≤ ε µ -a.e.Therefore, from Lemma 2.13 it follows thatlim sup m →∞ Z Ω g m dµ ≤ Z Ω lim sup m →∞ g m dµ ≤ ε, as required. (cid:3) Let (Ω , Σ , µ ) be a probability space and X be a Banach space. Recallthat a function ϕ : Ω → X ∗ is said to be w ∗ -scalarly µ -measurable if forevery x ∈ X the composition h x, ϕ ( · ) i : Ω → R is µ -measurable. It isknown that any element of L ( µ, X ) ∗ can be identified with a w ∗ -scalarly µ -measurable function ϕ : Ω → X ∗ in such a way that k ϕ ( · ) k X ∗ ∈ L ∞ ( µ )and k ϕ k L ( µ,X ) ∗ = kk ϕ ( · ) k X ∗ k L ∞ ( µ ) , the duality being h h, ϕ i = Z Ω h h ( · ) , ϕ ( · ) i dµ for all h ∈ L ( µ, X )(see, e.g., [10, Theorem 1.5.4]). We will use this representation of L ( µ, X ) ∗ in the proofs of Lemmas 2.17 and 2.18 below. Notation 2.15.
Given x ∈ X and f ∈ L ( µ ), we write f ⊗ x ∈ L ( µ, X )to denote the (equivalence class of the) function defined by( f ⊗ x )( ω ) := f ( ω ) x for µ -a.e. ω ∈ Ω.Observe that k f ⊗ x k L ( µ,X ) = k f k L ( µ ) k x k X .Throughout the rest of this subsection the unit interval [0 ,
1] is equippedwith the Lebesgue measure and we denote by ( r n ) n ∈ N the sequence ofRademacher functions on [0 , Lemma 2.16.
Let Z be a Banach space and ( z n ) n ∈ N be an ℓ -sequence in Z with constant C > . Then ( r n ⊗ z n ) n ∈ N is an ℓ -sequence in L ([0 , , Z ) with constant C and so clust L ([0 , ,Z ) ∗∗ ( r n ⊗ z n ) εB L ([0 , ,Z ) ∗∗ for any ≤ ε < C .Proof. The first statement follows from a simple computation (see, e.g.,the proof of Proposition 2.2.1 in [10]). We have δ ( r n ⊗ z n ) ≥ C by [23,Lemma 5(i)], which clearly implies the second statement. (cid:3) Lemma 2.17.
Let Z be a Banach space, M ⊆ Z and ε ≥ . If M is ε -weakly precompact, then for every sequence ( z n ) n ∈ N in M we have clust L ([0 , ,Z ) ∗∗ ( r n ⊗ z n ) ⊆ εB L ([0 , ,Z ) ∗∗ . Proof.
Note that M is bounded and so ( r n ⊗ z n ) n ∈ N is bounded. Fix anarbitrary F ∈ clust L ([0 , ,Z ) ∗∗ ( r n ⊗ z n ). We claim that (cid:12)(cid:12) h F, ϕ i (cid:12)(cid:12) ≤ ε for every ϕ ∈ B L ([0 , ,Z ) ∗ .Indeed, take any ϕ ∈ B L ([0 , ,Z ) ∗ (represented as in the paragraph precedingNotation 2.15). Let ( r n k ⊗ z n k ) k ∈ N be a subsequence such that h r n k ⊗ z n k , ϕ i → h F, ϕ i as k → ∞ . -WEAKLY PRECOMPACT SETS IN BANACH SPACES 13 Since M is ε -weakly precompact, by passing to a further subsequence wecan assume that δ ( z n k ) ≤ ε . Write h := lim inf k →∞ h z n k , ϕ ( · ) i ∈ L ∞ [0 , k →∞ h z n k , ϕ ( t ) i − lim inf k →∞ h z n k , ϕ ( t ) i ≤ δ ( z n k ) ≤ ε for a.e. t ∈ [0 , , we can apply Lemma 2.14 to get(2.2) lim sup k →∞ Z (cid:12)(cid:12) h z n k , ϕ ( t ) i − h ( t ) (cid:12)(cid:12) dt ≤ ε. For each k ∈ N we have h r n k ⊗ z n k , ϕ i = Z r n k ( t ) h z n k , ϕ ( t ) i dt and so (cid:12)(cid:12) h r n k ⊗ z n k , ϕ i (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z r n k ( t ) h ( t ) dt + Z r n k ( t ) (cid:0) h z n k , ϕ ( t ) i − h ( t ) (cid:1) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z r n k ( t ) h ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) + Z (cid:12)(cid:12) h z n k , ϕ ( t ) i − h ( t ) (cid:12)(cid:12) dt. This inequality, (2.2) and the fact that ( r n k ) k ∈ N is weakly null in L [0 , |h F, ϕ i| ≤ ε . The proof is finished. (cid:3) Lemma 2.18.
Let (Ω , Σ , µ ) be a probability space, Y be a Banach space,and ε ≥ . Let ( h n ) n ∈ N be a sequence in L ∞ ( µ, Y ) such that sup n ∈ N k h n k L ∞ ( µ,Y ) < ∞ and clust Y ∗∗ ( h n ( ω )) ⊆ εB Y ∗∗ for µ -a.e. ω ∈ Ω .Then clust L ( µ,Y ) ∗∗ ( h n ) ⊆ εB L ( µ,Y ) ∗∗ . Proof.
Note that ( h n ) n ∈ N is bounded in L ( µ, X ). Fix H ∈ clust L ( µ,Y ) ∗∗ ( h n ).Take any ϕ ∈ B L ( µ,Y ) ∗ (represented as in the paragraph preceding Nota-tion 2.15). For each n ∈ N we define g n ∈ L ∞ ( µ ) by g n := |h h n ( · ) , ϕ ( · ) i| .By the assumptions, we have sup n ∈ N k g n k L ∞ ( µ ) < ∞ and lim sup n →∞ g n ≤ εµ -a.e. Now, we can apply Lemma 2.13 to getlim sup n →∞ Z Ω g n dµ ≤ Z Ω lim sup n →∞ g n dµ ≤ ε, and so (cid:12)(cid:12) h H, ϕ i (cid:12)(cid:12) ≤ lim sup n →∞ (cid:12)(cid:12) h h n , ϕ i (cid:12)(cid:12) = lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z Ω h h n ( · ) , ϕ ( · ) i dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup n →∞ Z Ω g n dµ ≤ ε. This shows that clust L ( µ,Y ) ∗∗ ( h n ) ⊆ εB L ( µ,Y ) ∗∗ . (cid:3) The following lemma belongs to the folklore. We include a proof since wedid not find any suitable reference for it.
Lemma 2.19.
Let (Ω , Σ , µ ) be a probability space, X be a Banach space, f : Ω → X be a strongly µ -measurable function and g ∈ L [0 , . Define h g,f : Ω → L ([0 , , X ) by h g,f ( ω ) := g ⊗ f ( ω ) for all ω ∈ Ω . Then: (i) h g,f is strongly µ -measurable; (ii) h g,f is µ -essentially bounded whenever f is µ -essentially bounded.Proof. (i) Clearly, h g,f is a simple function whenever f is. In the generalcase, if f n : Ω → X is a sequence of simple functions converging to f µ -a.e.,then ( h g,f n ) n ∈ N is a sequence of simple functions converging to h g,f µ -a.e.,because k h g,f n ( ω ) − h g,f ( ω ) k L ([0 , ,X ) = k g ⊗ ( f n ( ω ) − f ( ω )) k L ([0 , ,X ) = k g k L [0 , k f n ( ω ) − f ( ω ) k X for every ω ∈ Ω and for every n ∈ N . Thus, h g,f is strongly µ -measurable.(ii) This is immediate from (i) and the equality k h g,f ( ω ) k L ([0 , ,X ) = k g k L [0 , k f ( ω ) k X which holds for every ω ∈ Ω. (cid:3) We also isolate for easy reference the following standard fact, which fol-lows from Chebyshev’s inequality.
Lemma 2.20.
Let (Ω , Σ , µ ) be a probability space and X be a Banach space.If W ⊆ L ( µ, X ) is uniformly integrable, then for every η > there is ρ > such that W ⊆ L ( ρB X ) + ηB L ( µ,X ) . More precisely, for every f ∈ W there is A ∈ Σ such that f χ A ∈ L ( ρB X ) and k f χ Ω \ A k L ( µ,X ) ≤ η . We have already gathered all the tools needed to prove Theorem 2.11. -WEAKLY PRECOMPACT SETS IN BANACH SPACES 15
Proof of Theorem 2.11. (ii) follows from (i), Theorem 2.1(i) and Lemma 2.5.We begin the proof of (i) with the following:
Particular case. Suppose that f n ∈ L ∞ ( µ, X ) for all n ∈ N and that sup n ∈ N k f n k L ∞ ( µ,X ) < ∞ . Write Y := L ([0 , , X ). For each n ∈ N we define (thanks to Lemma 2.19) h n ∈ L ∞ ( µ, Y ) by h n ( ω ) := r n ⊗ f n ( ω ) for µ -a.e. ω ∈ Ω.By the assumption on F and Lemma 2.17, we haveclust Y ∗∗ ( h n ( ω )) ⊆ εB Y ∗∗ for µ -a.e. ω ∈ Ω.Bearing in mind that sup n ∈ N k h n k L ∞ ( µ,Y ) < ∞ , an appeal to Lemma 2.18ensures that clust L ( µ,Y ) ∗∗ ( h n ) ⊆ εB L ( µ,Y ) ∗∗ . Write Z := L ( µ, X ) and let Φ : L ( µ, Y ) → L ([0 , , Z ) be the naturalisometric isomorphism. Then Φ( h n ) = r n ⊗ f n for all n ∈ N and thereforeclust L ([0 , ,Z ) ∗∗ ( r n ⊗ f n ) ⊆ εB L ([0 , ,Z ) ∗∗ . Therefore, ( f n ) n ∈ N cannot be an ℓ -sequence with constant C > ε (byLemma 2.16), as required.We now turn to the:
General case.
Fix
C > ε and choose any 0 < η < C − ε . Since ( f n ) n ∈ N isuniformly integrable, Lemma 2.20 gives a sequence ( A n ) n ∈ N in Σ such thateach g n := f n χ A n belongs to L ∞ ( µ, X ) and: • sup n ∈ N k g n k L ∞ ( µ,X ) < ∞ , • k f n − g n k L ( µ,X ) ≤ η for every n ∈ N .Define a multi-function ˜ F : Ω → P ( X ) by ˜ F ( ω ) := F ( ω ) ∪ { } for every ω ∈ Ω. By the
Particular case , the sequence ( g n ) n ∈ N in S ( ˜ F ) cannot be an ℓ -sequence with constant C − η . Since k f n − g n k L ( µ,X ) ≤ η for all n ∈ N ,we conclude that ( f n ) n ∈ N cannot be an ℓ -sequence with constant C . Theproof is finished. (cid:3) Question 2.21.
Is constant optimal in Theorem 2.11? The concept of δ C -sets in Lebesgue-Bochner spaces (which correspondsto the case ε = 0 of the following definition) appeared first in [8] and wasdiscussed further in [6]. Definition 2.22.
Let (Ω , Σ , µ ) be a probability space, X be a Banach space, W ⊆ L ( µ, X ) , and ε ≥ . We say that W is a δ C ε -set if it is uniformlyintegrable and for each δ > there is an ε -weakly precompact set M ⊆ X such that for every f ∈ W there is A ∈ Σ (depending on f ) with µ ( A ) ≥ − δ such that f ( ω ) ∈ M for µ -a.e. ω ∈ A . If in addition ε = 0 , then we justsay that W is a δ C -set . Every δ C -set is weakly precompact, but the converse does not hold ingeneral, see [8] and [6, Example 3]. It is known that uniform integrabilityand being a δ C -set are equivalent properties if X does not contain subspacesisomorphic to ℓ (see [8, Corollary 9 and Theorem 14]).The last result of this section is an extension of Corollary 2.12. To dealwith it we need a lemma that will also be used later. We omit its straight-forward proof. Lemma 2.23.
Let X be a Banach space. (i) If ( x n ) n ∈ N and ( y n ) n ∈ N are bounded sequences in X , then δ ( x n + y n ) ≤ δ ( x n ) + δ ( y n ) . (ii) If ε , ε ≥ and M i ⊆ X is ε i -weakly precompact for each i ∈ { , } ,then M + M is ( ε + ε ) -weakly precompact. (iii) If ε ≥ and M ⊆ X is ε -weakly precompact, then M is ε -weaklyprecompact. Corollary 2.24.
Let (Ω , Σ , µ ) be a probability space, X be a Banach space,and ε ≥ . Then every δ C ε -set of L ( µ, X ) is ε -weakly precompact.Proof. Let W ⊆ L ( µ, X ) be a δ C ε -set. Fix η >
0. Choose δ > f ∈ W Z C k f ( · ) k X dµ ≤ η for every C ∈ Σ with µ ( C ) ≤ δ .Let M ⊆ X be an ε -weakly precompact set such that, for each f ∈ W ,there is A f ∈ Σ such that µ ( A f ) ≥ − δ and f ( ω ) ∈ M for µ -a.e. ω ∈ A f .Then for each f ∈ W we have f χ A f ∈ L ( M ∪ { } ) and k f χ Ω \ A f k L ( µ,X ) ≤ η ,hence(2.3) W ⊆ L ( M ∪ { } ) + ηB L ( µ,X ) . Since M ∪ { } is ε -weakly precompact, Corollary 2.12 ensures that theset L ( M ∪ { } ) is 2 ε -weakly precompact. By (2.3) and Lemma 2.23(ii), W is (2 ε + 2 η )-weakly precompact. As η > W is 2 ε -weaklyprecompact (Lemma 2.5). (cid:3) -WEAKLY PRECOMPACT SETS IN BANACH SPACES 17 Banach spaces having property KM w We begin this section by collecting some basic properties of Banach spaceshaving property KM w (Definition 1.4). The first result says somehow thatthis property can be handled by considering absolutely convex closed subsetsof the unit ball. Proposition 3.1.
Let X be a Banach space. The following statements areequivalent: (i) X has property KM w , i.e., there is a family { M n,p : n, p ∈ N } ofsubsets of X such that (a) M n,p is p -weakly precompact for all n, p ∈ N ; (b) for each weakly precompact set C ⊆ X and for each p ∈ N thereis n ∈ N such that C ⊆ M n,p . (ii) The same as (i) with each M n,p being absolutely convex and closed. (iii) There is a family { M n,p : n, p ∈ N } of subsets of B X such that (a) M n,p is p -weakly precompact for all n, p ∈ N ; (b) for each weakly precompact set C ⊆ B X and for each p ∈ N there is n ∈ N such that C ⊆ M n,p . (iv) The same as (iii) with each M n,p being absolutely convex and closed.Moreover, any of these families can be chosen such that M n,p ⊆ M n +1 ,p forall n, p ∈ N .Proof. The implications (ii) ⇒ (i) and (iv) ⇒ (iii) are obvious.(i) ⇒ (ii) and (iii) ⇒ (iv): It is easy to check that, for a given ε ≥
0, theunion of finitely many ε -weakly precompact subsets of a Banach space is ε -weakly precompact. So, if { M n,p : n, p ∈ N } is a family as in condition (i)(resp., (iii)) and we define ˜ M n,p := S ni =1 M i,p for all n, p ∈ N , then thefamily { ˜ M n,p : n, p ∈ N } also satisfies the requirements of condition (i)(resp., (iii)) and we have ˜ M n,p ⊆ ˜ M n +1 ,p for all n, p ∈ N . Each aco( ˜ M n,p ) is p -weakly precompact (by Theorem 2.3) and so the same holds for aco( ˜ M n,p )(Lemma 2.23(iii)). Thus, the family { aco( ˜ M n, p ) : n, p ∈ N } satisfies therequirements.(i) ⇒ (iii): Note that if { M n,p : n, p ∈ N } is a family as in condition (i),then { M n,p ∩ B X : n, p ∈ N } satisfies the requirements of (iii).(iii) ⇒ (i): Let φ : N → N × N be a bijection. Write φ ( n ) = ( φ ( n ) , φ ( n ))for all n ∈ N . Define ˜ M n,p := φ ( n ) M φ ( n ) ,pφ ( n ) for all n, p ∈ N . It is easy tocheck that the family { ˜ M n,p : n, p ∈ N } satisfies the required properties. (cid:3) Proposition 3.2.
Let X be a Banach space having property KM w . Thenevery subspace of X has property KM w .Proof. Let { M n,p : n, p ∈ N } be a family of subsets of X as in Definition 1.4.Then, for any subspace Y ⊆ X , the family { M n,p ∩ Y : n, p ∈ N } witnessesthat Y has property KM w . (cid:3) We next show that any SWPG Banach space has property KM w . Thisincludes all SWCG spaces (e.g., L ( µ ) for a finite measure µ , separable Schurspaces and, of course, reflexive spaces) but also non-SWCG spaces like c (note that every Banach space without isomorphic copies of ℓ is SWPG).As we will see later, any Banach space having property KM (Definition 1.3)also satisfies property KM w (Theorem 3.8). Proposition 3.3.
Let X be a Banach space. If X is SWPG, then it hasproperty KM w .Proof. Let C ⊆ X be a weakly precompact set such that, for every weaklyprecompact set C ⊆ X and for every ε >
0, there is n ∈ N such that C ⊆ nC + εB X . For each n ∈ N and for each p ∈ N we define M n,p := nC + 12 p B X , so that M n,p is p -weakly precompact (Lemma 2.23(ii)). Therefore, the fam-ily { M n,p : n, p ∈ N } witnesses that X has property KM w . (cid:3) Question 3.4.
Let X be a Banach space having property KM w . Is X isomorphic to a subspace of a SWPG space? Around weak sequential completeness.
It is known that everySWCG space is weakly sequentially complete (see [41, Theorem 2.5]). Moregenerally, every Banach space having property KM is weakly sequentiallycomplete (see [26, Theorem 2.20]). On the other hand, it is clear that if aBanach space is weakly sequentially complete and SWPG, then it is SWCG(cf. [29, Theorem 2.2]). We do not know whether properties KM and KM w are equivalent for weakly sequentially complete spaces but, as we will see inCorollary 3.10, this is indeed true for C -weakly sequentially complete spacesin the sense of [23, 24].The following lemma exploits the argument used to get the weak sequen-tial completeness of SWCG spaces in [41, Theorem 2.5]. A similar ideawas used in the proof of the weak sequential completeness of Banach spaceshaving property KM (see [26, Theorem 2.20]) and also in [29, Lemma 2.3]. -WEAKLY PRECOMPACT SETS IN BANACH SPACES 19 Lemma 3.5.
Let X be a Banach space and { M n,p : n, p ∈ N } be a familyof absolutely convex subsets of X such that M n,p ⊆ M n +1 ,p for all n, p ∈ N .Suppose that for each relatively weakly compact set L ⊆ X and for each p ∈ N there is n ∈ N such that L ⊆ M n,p . Then for each weakly precompactset H ⊆ X and for each p ∈ N there is n ∈ N such that H ⊆ M n,p .Proof. We argue by contradiction. Suppose there exist a weakly precompactset H ⊆ X and p ∈ N such that H M n,p for all n ∈ N . Let ( x n ) n ∈ N be a sequence in H such that x n M n,p for all n ∈ N . Since H isweakly precompact and M n,p ⊆ M n +1 ,p for every n ∈ N , by passing to asubsequence we can assume that ( x n ) n ∈ N is weakly Cauchy. Note that theset { n ∈ N : x n ∈ M m,p } is finite for every m ∈ N .For each n ∈ N and for each s ∈ { , } we define m s ( n ) := min n m ∈ N : x n ∈ sM m,p o , so that x n ∈ sM m,p if and only if m ≥ m s ( n ). Observe that m ( n ) ≤ m ( n )for all n ∈ N (because each M m,p is balanced and so M m,p ⊆ M m,p ). Let ψ : N → N be any function such that lim n →∞ nψ ( n ) = 0 and n ≤ ψ ( n ) for all n ∈ N .Observe that there is a subsequence ( x n k ) k ∈ N such that(3.1) ψ ( m ( n k )) < m ( n k +1 ) for all k ∈ N . Indeed, set n = 1 and suppose that n k ∈ N has already been chosen forsome k ∈ N . Since x n M ψ ( m ( n k )) ,p whenever n ≥ ψ ( m ( n k )), we canchoose n k +1 ∈ N with n k +1 > n k such that x n k +1 M ψ ( m ( n k )) ,p and so ψ ( m ( n k )) < m ( n k +1 ).On the other hand, since m ( n k ) ≤ ψ ( m ( n k )) (3.1) < m ( n k +1 ) ≤ m ( n k +1 ) for all k ∈ N , the sequence ( m ( n k )) k ∈ N is strictly increasing and so(3.2) lim k →∞ m ( n k ) ψ ( m ( n k )) = 0 . Define y k := x n k +1 − x n k for all k ∈ N , so that ( y k ) k ∈ N is a weakly nullsequence in X . By assumption, there is m ∈ N such that y k ∈ M m ,p forall k ∈ N , hence x n k +1 = y k + x n k ∈ M m ,p + M m ( n k ) ,p ⊆ M m ( n k ) ,p whenever m ( n k ) ≥ m (bear in mind that M m ( n k ) ,p is convex) and so m ( n k ) ψ ( m ( n k )) ≥ m ( n k +1 ) ψ ( m ( n k )) (3.1) > k . This contradicts (3.2) and finishes the proof. (cid:3) It turns out that property KM w can be characterized as follows: Proposition 3.6.
Let X be a Banach space. Then X has property KM w ifand only if there is a family { M n,p : n, p ∈ N } of subsets of X such that: (i) M n,p is p -weakly precompact for all n, p ∈ N ; (ii) for each relatively weakly compact set C ⊆ X and for each p ∈ N there is n ∈ N such that C ⊆ M n,p .Proof. The “only if” part is obvious. To prove the “if” part, note that foreach n ∈ N and for each p ∈ N the set S ni =1 M i, p is p -weakly precompactand therefore ˜ M n,p := aco n [ i =1 M i, p ! is p -weakly precompact (by Theorem 2.3), hence 2 ˜ M n,p is p -weakly pre-compact. Clearly, ˜ M n,p ⊆ ˜ M n +1 ,p for all n, p ∈ N . In addition, given anyrelatively weakly compact set C ⊆ X and p ∈ N , there is n ∈ N such that C ⊆ M n, p ⊆ ˜ M n,p . From Lemma 3.5 it follows that X has property KM w as witnessed by the family { M n,p : n, p ∈ N } . (cid:3) Lemma 3.7.
Let X be a Banach space, M ⊆ X and ε ≥ . If M is ε -relatively weakly compact, then M is ε -weakly precompact.Proof. It suffices to check that M is ε ′ -weakly precompact for every ε ′ > ε (Lemma 2.5). By contradiction, suppose that M is not ε ′ -weakly precom-pact. Then M contains an ℓ -sequence with constant ε ′ , say ( x n ) n ∈ N (byTheorem 2.1(i)). Hence k x ∗∗ − x k ≥ ε ′ > ε for every x ∗∗ ∈ clust X ∗∗ ( x n ) and for every x ∈ X ,see [23, Lemma 5(ii)]. This contradicts that M w ∗ ⊆ X + εB X ∗∗ . (cid:3) Theorem 3.8.
If a Banach space X has property KM , then it has prop-erty KM w . -WEAKLY PRECOMPACT SETS IN BANACH SPACES 21 Proof.
Let { M n,p : n, p ∈ N } be a family of subsets of X as in Defini-tion 1.3. Note that each M n,p is p -weakly precompact (by Lemma 3.7).Then { M n, p : n, p ∈ N } satisfies the conditions of Proposition 3.6, hence X has property KM w . (cid:3) Following [23, 24], a Banach space X is said to be C -weakly sequentiallycomplete for some C ≥ x n ) n ∈ N in X we haveinf x ∈ X k x ∗∗ − x k ≤ Cδ ( x n ) for every x ∗∗ ∈ clust X ∗∗ ( x n ).This quantitative version of weak sequential completeness appeared firstin [20], where it was shown that every L -embedded Banach space is 1-weaklysequentially complete (see [20, Lemma IV.7]). In fact, every L -embeddedBanach space is -weakly sequentially complete, see [23, Theorem 1]. Werefer the reader to [21, Chapter IV] for complete information on L -embeddedBanach spaces. Lemma 3.9.
Let X be a C -weakly sequentially complete Banach space forsome C ≥ . Let M ⊆ X and ε ′ > ε ≥ . If M is ε -weakly precompact,then it is Cε ′ -relatively weakly compact.Proof. There is nothing to prove if C = 0 (i.e., if X is reflexive), so weassume that C >
0. Suppose that the bounded set M is not 2 Cε ′ -relativelyweakly compact. Then for any 0 < η < C ( ε ′ − ε ) there is a sequence ( x n ) n ∈ N in M such that Cε ′ − η < inf {k x ∗∗ − x k : x ∗∗ ∈ clust X ∗∗ ( x n ) , x ∈ X } , see [2, Theorem 2.3]. Since X is C -weakly sequentially complete, we deducethat Cε < Cε ′ − η < Cδ ( x n k ) for every subsequence ( x n k ) k ∈ N . Hence M isnot ε -weakly precompact. (cid:3) Corollary 3.10.
Let X be a C -weakly sequentially complete Banach spacefor some C ≥ . Then X has property KM if and only if it has prop-erty KM w .Proof. The “only if” part follows from Theorem 3.8 and does not require theadditional assumption on X . Conversely, suppose that X has property KM w and let { M n,p : n, p ∈ N } be a family of subsets of X as in Definition 1.4.Choose q ∈ N with q > C . Observe that M n,qp is p -relatively weaklycompact for all n, p ∈ N , by Lemma 3.9. Therefore, { M n,qp : n, p ∈ N } satisfies the conditions of Definition 1.3 and X has property KM . (cid:3) Question 3.11.
Let X be a weakly sequentially complete Banach spacehaving property KM w . Does X have property KM ? Quotients.
In general, property KM w is not preserved by quotients.Indeed, every separable Banach space is a quotient of ℓ (which is SWCGand so it has property KM w ), but there are separable Banach spaces withoutproperty KM w , like C [0 ,
1] (see Corollary 3.26 in the next subsection). Themain result of this subsection is a three-space type result for property KM w ,namely: Theorem 3.12.
Let X be a Banach space and Y ⊆ X be a subspace notcontaining subspaces isomorphic to ℓ . Then X has property KM w if andonly if X/Y has property KM w . To deal with Theorem 3.12 we need a quantitative version of Mazur’stheorem on the existence of norm convergent convex block subsequences ofweakly convergent sequences. The following result is a particular case of [3,Theorem 4.1]:
Theorem 3.13.
Let Z be a Banach space, ( z j ) n ∈ N be a bounded sequencein Z , K ⊆ Z ∗ be a w ∗ -compact set, and a > . Suppose that for each z ∗ ∈ K there is j z ∗ ∈ N such that | z ∗ ( z j ) | ≤ a for every j ≥ j z ∗ . Then forevery ε > there is z ε ∈ co( { z j : j ∈ N } ) such that | z ∗ ( z ε ) | ≤ a + ε for all z ∗ ∈ K . Corollary 3.14.
Let Z be a Banach space, ( z j ) j ∈ N be a bounded sequencein Z , ( K n ) n ∈ N be a sequence of w ∗ -compact subsets of Z ∗ , and a n , η n > for all n ∈ N . Suppose that for each n ∈ N and for each z ∗ ∈ K n there is j n,z ∗ ∈ N such that | z ∗ ( z j ) | ≤ a n for every j ≥ j n,z ∗ . Then there is a convexblock subsequence ( y n ) n ∈ N of ( z j ) j ∈ N such that | z ∗ ( y n ) | ≤ a n + η n for every z ∗ ∈ K n and for every n ∈ N .Proof. We will construct the y n ’s inductively. For the first step, applyTheorem 3.13 (with K = K ) to get y ∈ co( { z j : j ∈ N } ) such that | z ∗ ( y ) | ≤ a + η for every z ∗ ∈ K . Write y = P j ∈ J a j z j for some finiteset J ⊆ N and some collection of non-negative real numbers ( a j ) j ∈ J with P j ∈ J a j = 1. Pick j ∈ N with j > max J . Now, Theorem 3.13 applied tothe tail ( z j ) j ≥ j and K = K ensures the existence of y ∈ co( { z j : j ≥ j } )such that | z ∗ ( y ) | ≤ a + η for every z ∗ ∈ K . By continuing in this waywe get the required convex block subsequence ( y n ) n ∈ N of ( z j ) j ∈ N . (cid:3) -WEAKLY PRECOMPACT SETS IN BANACH SPACES 23 Corollary 3.14 will be used in the proof of the following quantitativeversion of Lohman’s lifting [30] (cf. [9, 2.4.a]) as well as in Subsection 3.5.
Proposition 3.15.
Let X be a Banach space and Y ⊆ X be a subspacenot containing subspaces isomorphic to ℓ . Let q : X → X/Y be the quo-tient operator, M ⊆ X be a bounded set and ε ≥ . If q ( M ) is ε -weaklyprecompact, then M is ε -weakly precompact.Proof. Fix ε ′ > ε (to apply Lemma 2.5) and suppose that M is not ε ′ -weaklyprecompact. By Theorem 2.1(i), there is an ℓ -sequence with constant ε ′ contained in M , say ( x n ) n ∈ N . Since q ( M ) is ε -weakly precompact, by passingto a subsequence we can assume that δ ( q ( x n )) ≤ ε . Define z m := x m +1 − x m for all m ∈ N , so that ( z m ) m ∈ N is an ℓ -sequence with constant ε ′ .Fix any 0 < η < ε ′ − ε . Thensup ϕ ∈ B ( X/Y ) ∗ lim sup m →∞ | ϕ ( q ( z m )) | ≤ δ ( q ( x n )) < ε + η. Hence we can apply Corollary 3.14 to ( q ( z m )) m ∈ N (with K n = B ( X/Y ) ∗ forall n ∈ N ) to get a convex block subsequence (˜ z k ) k ∈ N of ( z m ) m ∈ N such that k q (˜ z k ) k X/Y < ε + η for every k ∈ N .Since ( z m ) m ∈ N is an ℓ -sequence with constant ε ′ , the same holds for(˜ z k ) k ∈ N (as it can be easily checked). Choose a sequence ( y k ) k ∈ N in Y suchthat k ˜ z k − y k k < ε + η for all k ∈ N . Then ( y k ) k ∈ N is an ℓ -sequence (withconstant ε ′ − ε − η ), which contradicts the fact that Y contains no subspaceisomorphic to ℓ . (cid:3) Proof of Theorem 3.12.
Let q : X → X/Y be the quotient operator. Sup-pose first that X has property KM w and let { M n,p : n, p ∈ N } be a familyof subsets of X satisfying the conditions of Definition 1.4. By Lemma 2.7,each q ( M n,p ) is p -weakly precompact. Let C ⊆ X/Y be a weakly precom-pact set. Choose a bounded set L ⊆ X such that q ( L ) = C . Then L isweakly precompact by Lohman’s lifting (i.e., Proposition 3.15 with ε = 0).Therefore, for every p ∈ N there is n ∈ N such that L ⊆ M n,p , and so C ⊆ q ( M n,p ). This shows that X/Y has property KM w , as witnessed bythe family { q ( M n,p ) : n, p ∈ N } .Conversely, suppose X/Y has property KM w and let { ˜ M n,p : n, p ∈ N } be a family of subsets of X/Y satisfying the conditions of Definition 1.4.We can assume that ˜ M n,p ⊆ ˜ M n +1 ,p for all n, p ∈ N (Proposition 3.1).Take bounded sets M n,p ⊆ X in such a way that q ( M n,p ) = ˜ M n,p and M n,p ⊆ M n +1 ,p for all n, p ∈ N . By Proposition 3.15, each M n,p is p -weaklyprecompact and so M ′ n,p := M n,p + nB Y ⊆ X is p -weakly precompact (Lemma 2.23(ii)).We claim that the family { M ′ n,p : n, p ∈ N } witnesses that X has prop-erty KM w . Indeed, let L ⊆ X be a weakly precompact set and fix p ∈ N .Since q ( L ) is weakly precompact in X/Y , we have q ( L ) ⊆ ˜ M n ,p = q ( M n ,p )for some n ∈ N . Now, if we choose n ∈ N large enough such that n ≥ n and n ≥ k x k X for all x ∈ L − M n ,p , then we have L ⊆ M n,p + nB Y = M ′ n,p . (cid:3) A similar argument using Lohman’s lifting and the fact that weak pre-compactness is preserved by taking absolutely convex hulls (Subsection 2.1)yields the following result. We omit the details.
Theorem 3.16.
Let X be a Banach space and Y ⊆ X be a subspace notcontaining subspaces isomorphic to ℓ . Then X is SWPG if and only if X/Y is SWPG.
Question 3.17.
Let X be a Banach space and Y ⊆ X be a subspace. (i) Does X have property KM w if both Y and X/Y have property KM w ? (ii) Is X SWPG if both Y and X/Y are SWPG?
We stress that it is also unknown whether the property of being SWCGis a three-space property. There is a result analogous to Theorem 3.16 forthe property of being SWCG when Y is reflexive, see [41, Theorem 2.7].3.3. Unconditional sums.
In this subsection we discuss the stability ofproperty KM w under unconditional sums.Given a sequence ( X m ) m ∈ N of Banach spaces, we denote by (cid:0)L m ∈ N X m (cid:1) ℓ ∞ its ℓ ∞ -sum. If E is a Banach space with a normalized 1-unconditionalbasis ( e m ) m ∈ N , we write (cid:0)L m ∈ N X m (cid:1) E to denote the E -sum of ( X m ) m ∈ N ,that is, the Banach space M m ∈ N X m ! E := ( ( x m ) m ∈ N ∈ Y m ∈ N X m : X m ∈ N k x m k e m converges in E ) equipped with the norm k ( x m ) m ∈ N k ( L m ∈ N X m ) E := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X m ∈ N k x m k e m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E . The simplest examples are obtained when E is c or ℓ p for 1 ≤ p < ∞ (equipped with its usual basis). -WEAKLY PRECOMPACT SETS IN BANACH SPACES 25 The fact that the ℓ -sum of countably many SWPG spaces is SWPG wasstated without proof in [27, Example 4.1(c)]. We include a proof below(Proposition 3.19) for the sake of completeness and also because a simi-lar argument yields the analogue for Banach spaces having property KM w (Proposition 3.20). The key is condition (∆) of the following folk lemma. Lemma 3.18.
Let ( X m ) m ∈ N be a sequence of Banach spaces and let us write X := ( L m ∈ N X m ) ℓ . Let π m : X → X m be the canonical projection for all m ∈ N . Then a set C ⊆ X is weakly precompact if and only if π m ( C ) isweakly precompact in X m for all m ∈ N and the following condition holds: (∆) for every δ > there is m ∈ N such that X m>m k π m ( x ) k X m ≤ δ for all x ∈ C .Proof. The “if” part follows easily from Lemma 2.23(ii) and the fact that X ∗ can be identified with (cid:0)L m ∈ N X ∗ m (cid:1) ℓ ∞ , the duality being (cid:10) ( x m ) m ∈ N , ( x ∗ m ) m ∈ N (cid:11) = X m ∈ N h x m , x ∗ m i for every ( x m ) m ∈ N ∈ X and for every ( x ∗ m ) m ∈ N ∈ X ∗ .To prove the “only if” part, suppose that C ⊆ X is weakly precompact.Clearly, π m ( C ) is weakly precompact in X m for all m ∈ N . For each n ∈ N and for each p ∈ N , we define M n,p := { x ∈ X : π m ( x ) = 0 for all m > n } + 12 p B X . Since (∆) holds for any relatively weakly compact subset of X (see, e.g.,[22, Lemma 7.2(ii)]), the family { M n,p : n, p ∈ N } satisfies the condi-tions of Lemma 3.5. Fix δ > p ∈ N such that p ≤ δ . ByLemma 3.5, there is m ∈ N such that C ⊆ M m ,p , which implies that P m>m k π m ( x ) k X m ≤ δ for all x ∈ C . (cid:3) Proposition 3.19.
Let ( X m ) m ∈ N be a sequence of SWPG Banach spaces.Then ( L m ∈ N X m ) ℓ is SWPG.Proof. Write X := ( L m ∈ N X m ) ℓ . For each m ∈ N we fix a weakly pre-compact set C m ⊆ m B X m with the following property: for every weaklyprecompact set H ⊆ X m and for every ε > n ∈ N such that H ⊆ nC m + εB X m . Since weak precompactness is preserved by taking ab-solutely convex hulls (see Subsection 2.1), we can assume that each C m is absolutely convex. Define W := Y m ∈ N C m ⊆ X, so that W is weakly precompact in X (by Lemma 3.18). Now, take anyweakly precompact set C ⊆ X and ε >
0. Lemma 3.18 applied to C ensuresthe existence of m ∈ N such that(3.3) X m>m k π m ( x ) k X m ≤ ε x ∈ C .For each m ∈ { , . . . , m } the set π m ( C ) ⊆ X m is weakly precompact andso there is n m ∈ N such that π m ( C ) ⊆ n m C m + ε m B X m . If we write n := max { n , . . . , n m } , then(3.4) π m ( C ) ⊆ nC m + ε m B X m for every m ∈ { , . . . , m } (because each C m is balanced). From (3.3) and (3.4) we get C ⊆ nW + εB X .This shows that X is SWPG. (cid:3) Proposition 3.20.
Let ( X m ) m ∈ N be a sequence of Banach spaces havingproperty KM w . Then ( L m ∈ N X m ) ℓ has property KM w .Proof. Write X := ( L m ∈ N X m ) ℓ . Fix m ∈ N . Let us take a family { M mn,p : n, p ∈ N } of subsets of X m satisfying the conditions of Definition 1.4,with the additional property that M mn,p ⊆ M mn +1 ,p for all n, p ∈ N (Propo-sition 3.1). If we denote by j m : X m → X the canonical embedding, theneach ˜ M mn,p := j m ( M mn,p ) is p -weakly precompact in X (apply Lemma 2.7).Let φ : N → N × N be a bijection and write φ ( n ) = ( φ ( n ) , φ ( n )) for all n ∈ N . For each n ∈ N and for each p ∈ N , the set M n,p := φ ( n ) X m =1 ˜ M mφ ( n ) , pφ ( n ) + 14 p B X is p -weakly precompact in X (apply Lemma 2.23(ii)). Fix a weakly pre-compact set C ⊆ X and take any p ∈ N . By Lemma 3.18, there is m ∈ N such that(3.5) X m>m k π m ( x ) k X m ≤ p for all x ∈ C ,where π m : X → X m is the canonical projection. For each m ∈ { , . . . , m } the set π m ( C ) is weakly precompact, hence π m ( C ) ⊆ M mk m , pm for some k m ∈ N . Write k := max { k , . . . , k m } , so that π m ( C ) ⊆ M mk, pm for every -WEAKLY PRECOMPACT SETS IN BANACH SPACES 27 m ∈ { , . . . , m } . Take n ∈ N such that φ ( n ) = ( φ ( n ) , φ ( n )) = ( k, m ).Then C (3.5) ⊆ m X m =1 j m ( π m ( C )) + 14 p B X ⊆ m X m =1 ˜ M mk, pm + 14 p B X = M n,p . This shows that the family { M n,p : n, p ∈ N } fulfills the conditions ofDefinition 1.4 and so X has property KM w . (cid:3) The situation changes for ℓ p -sums when 1 < p < ∞ . Indeed, if ( X m ) m ∈ N isa sequence of Banach spaces and ( L m ∈ N X m ) ℓ p is SWPG (or just a subspaceof a SWPG space), then X m contains no subspace isomorphic to ℓ for allbut finitely many m ∈ N , see [27, Theorem 4.5] (resp., [29, Theorem 2.6]).In particular, for 1 < p < ∞ the space ℓ p ( ℓ ) does not embed isomorphicallyinto a SWPG space. Theorem 3.23 below uses similar ideas to extend thoseresults to property KM w and more general unconditional sums.The following well-known lemma provides a useful representation for thedual of an unconditional sum. Lemma 3.21.
Let ( X m ) m ∈ N be a sequence of Banach spaces, E be a Banachspace with a normalized -unconditional basis ( e m ) m ∈ N , and let us write X := (cid:0)L m ∈ N X m (cid:1) E to denote the corresponding E -sum. Let ( e ∗ m ) m ∈ N be thesequence in E ∗ of biorthogonal functionals associated with ( e m ) m ∈ N . Then: (i) For every x ∗ = ( x ∗ m ) m ∈ N ∈ Q m ∈ N X ∗ m satisfying ||| x ∗ ||| := sup M ∈ N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M X m =1 k x ∗ m k e ∗ m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ∗ < ∞ we can define ϕ x ∗ ∈ X ∗ by ϕ x ∗ (( x m ) m ∈ N ) := X m ∈ N x ∗ m ( x m ) for all ( x m ) m ∈ N ∈ X .Moreover, k ϕ x ∗ k X ∗ = ||| x ∗ ||| . (ii) For every ϕ ∈ X ∗ there is x ∗ ∈ Q m ∈ N X ∗ m with ||| x ∗ ||| < ∞ suchthat ϕ = ϕ x ∗ . Lemma 3.22.
Let ( X m ) m ∈ N be a sequence of Banach spaces and E be aBanach space with a normalized -unconditional basis ( e m ) m ∈ N such that E ∗ is separable. Let X := (cid:0)L m ∈ N X m (cid:1) E be the corresponding E -sum and π m : X → X m be the canonical projection for all m ∈ N . Then a sequence ( y j ) j ∈ N in X is weakly null if and only if it is bounded and the sequence ( π m ( y j )) j ∈ N is weakly null in X m for every m ∈ N . Proof.
The “only if” part is immediate and does not require the assumptionthat E ∗ is separable.Let us prove the “if” part. We follow the notations of Lemma 3.21.Fix ϕ ∈ X ∗ and write ϕ = ϕ x ∗ for some x ∗ = ( x ∗ m ) m ∈ N ∈ Q m ∈ N X ∗ m with ||| x ∗ ||| < ∞ . Since E ∗ is separable, ( e ∗ m ) m ∈ N is a normalized 1-unconditionalboundedly-complete basis of E ∗ (see, e.g., [1, Theorems 3.2.12 and 3.3.1]),hence the series P m ∈ N k x ∗ m k e ∗ m converges unconditionally in E ∗ . Take any ε >
0. Choose M ∈ N large enough such that (cid:13)(cid:13)P m ∈ F k x ∗ m k e ∗ m (cid:13)(cid:13) E ∗ ≤ ε C forevery finite set F ⊆ N \ { , . . . , M } , where C > k y j k X ≤ C for all j ∈ N . Consider the elements x ∗ := ( x ∗ , . . . , x ∗ M , , , . . . ) and x ∗ := (0 , . . . , | {z } M times , x ∗ M +1 , x ∗ M +2 , . . . )of Q m ∈ N X ∗ m . Then ||| x ∗ ||| ≤ ε C and ϕ = ϕ x ∗ + ϕ x ∗ , so there is j ∈ N suchthat | ϕ ( y j ) | ≤ | ϕ x ∗ ( y j ) | + | ϕ x ∗ ( y j ) | ≤ M X m =1 (cid:12)(cid:12) x ∗ m (cid:0) π m ( y j ) (cid:1)(cid:12)(cid:12) + ε ≤ ε for every j ≥ j (each ( π m ( y j )) j ∈ N is weakly null in X m ). It follows that( ϕ ( y j )) j ∈ N converges to 0. As ϕ ∈ X ∗ is arbitrary, ( y j ) j ∈ N is weakly null. (cid:3) Theorem 3.23.
Let ( X m ) m ∈ N be a sequence of Banach spaces and E bea Banach space with a normalized -unconditional basis such that E ∗ isseparable. If the E -sum X := ( L m ∈ N X m ) E has property KM w , then X m contains no subspace isomorphic to ℓ for all but finitely many m ∈ N .Proof. Since property KM w in inherited by subspaces (Proposition 3.2), itsuffices to prove that if X m contains a subspace isomorphic to ℓ for every m ∈ N , then X fails property KM w . So, we assume that each X m containsa subspace isomorphic to ℓ . By James’ ℓ -distortion theorem (see, e.g.,[1, Theorem 10.3.1]), for each m ∈ N there is a normalized ℓ -sequencewith constant in X m , say ( x mk ) k ∈ N . Suppose, by contradiction, that X has property KM w . Fix a family { M n,p : n, p ∈ N } of subsets of X as inDefinition 1.4.Let Φ ⊆ N N be the set of all strictly increasing sequences in N . Fix ϕ ∈ Φ.For each j ∈ N , define y ϕ,j := (0 , . . . , , x jϕ ( j ) |{z} j th-term , , . . . ) ∈ X. -WEAKLY PRECOMPACT SETS IN BANACH SPACES 29 The sequence ( y ϕ,j ) j ∈ N is weakly null in X (by Lemma 3.22). Therefore,the set K ϕ := { y ϕ,j : j ∈ N } is relatively weakly compact in X . Hence K ϕ ⊆ M n ( ϕ ) , for some n ( ϕ ) ∈ N .We claim that there is n ∈ N such that A n := { ϕ ( n ) : ϕ ∈ Φ , n ( ϕ ) = n } is infinite. Indeed, otherwise we can find ϕ ∈ Φ such that ϕ ( n ) > max { ϕ ( n ) : ϕ ∈ Φ , n ( ϕ ) = n } for all n ∈ N ,which leads to a contradiction when n = n ( ϕ ).Enumerate A n = { ϕ k ( n ) : k ∈ N } for some sequence ( ϕ k ) k ∈ N in Φ with n ( ϕ k ) = n for all k ∈ N . Define z k := y ϕ k ,n ∈ K ϕ k ⊆ M n , for all k ∈ N .Observe that each z k has norm 1 and that for every s ∈ N and for all a , . . . , a s ∈ R we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s X k =1 a k z k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s X k =1 a k x n ϕ k ( n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X n ≥ s X k =1 | a k | . It follows that M n , contains an ℓ -sequence with constant , which con-tradicts the fact that M n , is -weakly precompact (Theorem 2.1(ii)). Theproof is finished. (cid:3) The following corollary extends the corresponding result for ℓ p -sums and1 < p < ∞ , which was proved in [26, Corollary 2.28]. Corollary 3.24.
Let ( X m ) m ∈ N be a sequence of Banach spaces and E bea Banach space with a normalized -unconditional basis such that E ∗ isseparable. If the E -sum ( L m ∈ N X m ) E has property KM , then X m is reflexivefor all but finitely many m ∈ N .Proof. On the one hand, since property KM implies weak sequential com-pleteness (see [26, Theorem 2.20]), each X m is weakly sequentially complete.On the other hand, by Theorems 3.8 and 3.23, X m contains no subspaceisomorphic to ℓ for all but finitely many m ∈ N . Now, the conclusionfollows from the fact that every weakly sequentially complete Banach spacenot containing subspaces isomorphic to ℓ is reflexive. (cid:3) Corollary 3.25.
The spaces c ( ℓ ) and ℓ p ( ℓ ) for < p < ∞ fail prop-erty KM w . The space C [0 ,
1] contains an isometric copy of any separable Banachspace, so from the previous corollary and Proposition 3.2 we get:
Corollary 3.26.
The space C [0 , fails property KM w . The Banach space of Batt and Hiermeyer [6, §
3] (which we will denoteby X BH ) was the first example of a weakly sequentially complete spacewhich is not SWCG (see [41, Example 2.6]). Simpler examples like ℓ ( ℓ )were given later (see [42, Theorem 5.10]). It is known that X BH also failsproperty KM (see [31, Example 2.9] and [26, Theorem 2.18]). We will showthat, in fact, X BH fails property KM w , because it contains an isometriccopy of ℓ ( ℓ ) (Corollary 3.28).The space X BH can be seen as a member of a class of Banach spaces builton adequate families which goes back to Kutzarova and Troyanski [28] (see,e.g., [4]). Recall that a family A of subsets of a non-empty set Γ is said tobe adequate if it satisfies the following conditions:(i) If A ∈ A and B ⊆ A , then B ∈ A .(ii) { γ } ∈ A for all γ ∈ Γ.(iii) If A ⊆ Γ and every finite subset of A belongs to A , then A ∈ A .For each function x : Γ → R , we define k x k E , ( A ) := sup p X i =1 X γ ∈ C i | x ( γ ) | ! / ∈ [0 , ∞ ] , where the supremum runs over all finite collections C , . . . , C p of pairwisedisjoint finite elements of A . The Banach space E , ( A ) is E , ( A ) := (cid:8) x ∈ R Γ : k x k E , ( A ) < ∞ (cid:9) , equipped with the pointwise operations and the norm k · k E , ( A ) . Clearly, E , ( A ) contains the linear space c (Γ) of all finitely supported real-valuedfunctions on Γ. For each γ ∈ Γ, let e γ ∈ c (Γ) be defined by e γ ( γ ′ ) := 0 forall γ ′ = γ , and e γ ( γ ) := 1. Proposition 3.27.
Let A be an adequate family of subsets of a non-emptyset Γ . Suppose there is a sequence ( A n ) n ∈ N in A such that each A n is infiniteand |{ n ∈ N : A n ∩ A = ∅}| ≤ for every A ∈ A .Then ℓ ( ℓ ) is isometrically isomorphic to a subspace of E , ( A ) .Proof. Observe that the A n ’s are pairwise disjoint. Choose a countableinfinite set { γ n,m : n, m ∈ N } ⊆ Γ in such a way that A n ⊇ { γ n,m : m ∈ N } -WEAKLY PRECOMPACT SETS IN BANACH SPACES 31 for all n ∈ N . Fix N, M ∈ N and a n,m ∈ R for all 1 ≤ n ≤ M and for all1 ≤ m ≤ M . We will prove that(3.6) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X n =1 M X m =1 a n,m e γ n,m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E , ( A ) = N X n =1 M X m =1 | a n,m | ! / . Clearly, this implies that the space ℓ ( ℓ ) is isometrically isomorphic to thesubspace span { e γ n,m : n, m ∈ N } ⊆ E , ( A ).Write x := P Nn =1 P Mm =1 a n,m e γ n,m andΓ := { γ n,m : 1 ≤ n ≤ N, ≤ m ≤ M } . Define ˜ A n := { γ n,m : 1 ≤ m ≤ M } for every 1 ≤ n ≤ N . Observe thateach ˜ A n belongs to A (because ˜ A n ⊆ A n ∈ A ) and ˜ A n ∩ ˜ A n ′ = ∅ whenever n = n ′ , so k x k E , ( A ) ≥ N X n =1 X γ ∈ ˜ A n | x ( γ ) | / = N X n =1 M X m =1 | a n,m | ! / . On the other hand, fix any finite collection C , . . . , C p of pairwise disjointfinite elements of A . For each 1 ≤ n ≤ N , we write I n := { ≤ i ≤ p : ˜ A n ∩ C i = ∅} . Observe that the I n ’s are pairwise disjoint and that Γ ∩ C i = ˜ A n ∩ C i forevery i ∈ I n and for every 1 ≤ n ≤ N . Then p X i =1 X γ ∈ C i | x ( γ ) | ! = p X i =1 X γ ∈ Γ ∩ C i | x ( γ ) | ! = N X n =1 X i ∈ I n X γ ∈ ˜ A n ∩ C i | x ( γ ) | ≤ N X n =1 X i ∈ I n X γ ∈ ˜ A n ∩ C i | x ( γ ) | = N X n =1 X γ ∈ ˜ A n ∩ ( S i ∈ In C i ) | x ( γ ) | ≤ N X n =1 M X m =1 | a n,m | ! . By the very definition of the norm in E , ( A ), it follows that k x k E , ( A ) ≤ N X n =1 M X m =1 | a n,m | ! / . This finishes the proof of (3.6). (cid:3)
The Batt-Hiermeyer space [6, §
3] is defined as X BH := E , ( A ) where A is the adequate family of all chains of the dyadic tree T , i.e., the set T := {∅} ∪ S n ∈ N { , } n of all finite sequences of 0’s and 1’s. By a chain of T we mean a set A ⊆ T such that for every σ, σ ′ ∈ A we have thateither σ extends σ ′ or vice versa. The space X BH is a separable, weaklysequentially complete, dual Banach space. Corollary 3.28.
The space X BH contains a subspace isometrically isomor-phic to ℓ ( ℓ ) . In particular, X BH fails property KM w .Proof. The last assertion follows from the first one, Corollary 3.25 andProposition 3.2. In order to prove the first assertion it is enough to checkthat the family of all chains of T satisfies the condition of Proposition 3.27.For each n ∈ N and for each m ∈ N we define σ n,m := (0 , , . . . , | {z } n times , , , . . . , | {z } m times ) ∈ T . Now, it is easy to check that the sequence of chains ( A n ) n ∈ N of T definedby A n := { σ n,m : m ∈ N } satisfies the required condition. (cid:3) Property KM w and Lebesgue-Bochner spaces. Given a proba-bility space (Ω , Σ , µ ) and a Banach space X , it is unknown whether theproperty of being SWPG lifts from X to L ( µ, X ). This is indeed the caseif X contains no subspace isomorphic to ℓ (see [29, Example 3.5(ii)]). As itwas pointed out in [29, Remark 3.4], a general positive answer would implythat the property of being SWCG lifts from X to L ( µ, X ), thus answeringa long standing open question of Schl¨uchtermann and Wheeler [41].In this subsection we address the same type of question for property KM w .Similarly, we do not know whether L ( µ, X ) has property KM w whenever X does. The following partial answer is similar to previous results for SWCGand SWPG spaces, see [35, Theorem 2.7] and [29, Proposition 3.8]. Itinvolves the notion of δ C -set which was recalled in Definition 2.22. Proposition 3.29.
Let (Ω , Σ , µ ) be a probability space and X be a Banachspace having property KM w . Then there is a family { ˜ M n,p : n, p ∈ N } ofsubsets of L ( µ, X ) such that: (i) ˜ M n,p is p -weakly precompact for all n, p ∈ N . -WEAKLY PRECOMPACT SETS IN BANACH SPACES 33 (ii) For each δ C -set W ⊆ L ( µ, X ) and for each p ∈ N there is n ∈ N such that W ⊆ ˜ M n,p .Proof. Let { M n,p : n, p ∈ N } be a family of subsets of X as in Definition 1.4.For each n ∈ N and for each p ∈ N , the set˜ M n,p := L ( M n, p ) + 14 p B L ( µ,X ) is p -weakly precompact in L ( µ, X ), by Corollary 2.12 and Lemma 2.23(ii).To check condition (ii), fix a δ C -set W ⊆ L ( µ, X ) and take any p ∈ N . Asin the proof of Corollary 2.24 (with ε = 0), there is a weakly precompactset C ⊆ X such that W ⊆ L ( C ) + p B L ( µ,X ) . Since C ⊆ M n, p for some n ∈ N , we get W ⊆ ˜ M n,p . (cid:3) Our next result strengthens the conclusion of the Bourgain-Maurey-Pisiertheorem (i.e., Theorem 2.11 with ε = 0) for Banach spaces having prop-erty KM w and multi-functions which are “measurable” in a certain sense.The proof is similar to that of [36, Proposition 2.8]. Proposition 3.30.
Let (Ω , Σ , µ ) be a probability space, X be a Banachspace having property KM w and F : Ω → P ( X ) be a multi-function suchthat: (i) F ( ω ) is weakly precompact for µ -a.e. ω ∈ Ω ; (ii) { ω ∈ Ω : F ( ω ) ⊆ C } ∈ Σ for every absolutely convex closed set C ⊆ X .Write S ( F ) := { f ∈ L ( µ, X ) : f ( ω ) ∈ F ( ω ) for µ -a.e. ω ∈ Ω } to denote the set of all (equivalence classes of ) Bochner µ -integrable selectorsof F . Then every uniformly integrable subset of S ( F ) is a δ C -set.Proof. It suffices to check that for every δ > W ⊆ X and A ∈ Σ with µ ( A ) ≥ − δ such that F ( ω ) ⊆ W forevery ω ∈ A .Let { M n,p : n, p ∈ N } be a family of absolutely convex closed subsetsof X witnessing property KM w and such that M n,p ⊆ M n +1 ,p for all n, p ∈ N (Proposition 3.1). Then A n,p := { ω ∈ Ω : F ( ω ) ⊆ M n,p } ∈ Σ for all n, p ∈ N . Given any p ∈ N , we have µ ( S n ∈ N A n,p ) = 1 (because F ( ω )is weakly precompact for µ -a.e. ω ∈ Ω) and A n,p ⊆ A n +1 ,p for every n ∈ N ,so there is n p ∈ N such that µ ( A n p ,p ) ≥ − δ p .Define A := T p ∈ N A n p ,p ∈ Σ, so that µ ( A ) ≥ − δ . Observe that F ( ω ) ⊆ W := \ p ∈ N M n p ,p for every ω ∈ A and that W is weakly precompact, because each M n p ,p is p -weakly precom-pact (and then we can apply Lemma 2.5). (cid:3) A remark on property (K).
A Banach space X is said to have prop-erty (K) if every w ∗ -convergent sequence in X ∗ admits a convex block sub-sequence which converges with respect to the Mackey topology µ ( X ∗ , X ),that is, the topology on X ∗ of uniform convergence on weakly compactsubsets of X . This concept is due to Kwapie´n and appeared first in [25].Property (K) and some related properties have also been studied in [5, 11,12, 17, 18, 37].Subspaces of SWCG spaces have property (K) (see [5, Corollary 2.3], cf.[12, Corollary 3.8]). The main result of this subsection generalizes thatstatement: Theorem 3.31.
If a Banach space X has property KM , then it has prop-erty (K). In the proof of Theorem 3.31 we will use the following quantitative ex-tension of [5, Lemma 2.11]:
Lemma 3.32.
Let X be a Banach space, ( x ∗ j ) j ∈ N be a w ∗ -null sequencein B X ∗ , and ε n > ε ≥ for all n ∈ N . If ( M n ) n ∈ N is a sequence of ε -relatively weakly compact subsets of X , then there is a convex block sub-sequence ( y ∗ n ) n ∈ N of ( x ∗ j ) j ∈ N such that | y ∗ n ( x ∗∗ ) | ≤ ε n for every x ∗∗ ∈ M nw ∗ and for every n ∈ N .Proof. Fix n ∈ N and choose η n > ε + 2 η n ≤ ε n . Observe thatfor every x ∗∗ ∈ M nw ∗ there is x ∈ X such that k x ∗∗ − x k ≤ ε , hence | x ∗ j ( x ∗∗ ) | ≤ ε + | x ∗ j ( x ) | ≤ ε + η n for j ∈ N large enough (depending on x ∗∗ ).So, we can apply Corollary 3.14 to the sequence ( x ∗ j ) j ∈ N (with Z = X ∗ , K n = M nw ∗ and a n = ε + η n for all n ∈ N ) to obtain the required convexblock subsequence. (cid:3) -WEAKLY PRECOMPACT SETS IN BANACH SPACES 35 Proof of Theorem 3.31.
Let { M n,p : n, p ∈ N } be a family of subsets of X satisfying the conditions of Definition 1.3. Observe that, for any ε ≥ ε -relatively weakly compact subsets of a Banachspace is ε -relatively weakly compact. So, we can assume without loss ofgenerality that M n,p ⊆ M n +1 ,p for every n ∈ N and for every p ∈ N .Fix a w ∗ -null sequence in B X ∗ (to check property (K) it suffices to considersuch sequences). We can apply inductively Lemma 3.32 to get, for each p ∈ N , a sequence ( x ∗ n,p ) n ∈ N in such a way that: • ( x ∗ n, ) n ∈ N is a convex block subsequence of ( x ∗ n ) n ∈ N ; • ( x ∗ n,p +1 ) n ∈ N is a convex block subsequence of ( x ∗ n,p ) n ∈ N for all p ∈ N ; • sup {| x ∗ n,p ( x ∗∗ ) | : x ∗∗ ∈ M n,pw ∗ } ≤ p + n for all n, p ∈ N .This inequality implies that, for each n ′ ≥ n in N and for each p ∈ N , wehave(3.7) sup x ∗∗ ∈ M n,pw ∗ | x ∗ ( x ∗∗ ) | ≤ p + 1 n ′ for every x ∗ ∈ co( { x ∗ m,p : m ≥ n ′ } ).Define ˜ x ∗ k := x ∗ k,k for every k ∈ N . Then (˜ x ∗ k ) k ∈ N is a convex blocksubsequence of ( x ∗ n ) n ∈ N . We claim that (˜ x ∗ k ) k ∈ N is µ ( X ∗ , X )-convergent to 0.Indeed, let C ⊆ X be a weakly compact set and fix ε >
0. Choose p ∈ N with p ≤ ε and then take n ∈ N such that C ⊆ M n ,p . Since (˜ x ∗ k ) k ≥ p is aconvex block subsequence of ( x ∗ n,p ) n ∈ N , inequality (3.7) yieldslim sup k →∞ sup x ∈ C | ˜ x ∗ k ( x ) | ≤ p ≤ ε. As ε > x ∗ k ) k ∈ N converges to 0 uniformlyon C . (cid:3) In [26, Theorem 2.18] it was pointed out that every Banach space X having property KM is a subspace of a weakly compactly generated space, asa consequence of [16, Theorem 1]; therefore, B X ∗ is w ∗ -sequentially compact(see, e.g., [13, p. 228, Theorem 4]). This fact and Theorem 3.31 give thefollowing result. Corollary 3.33.
Let X be a Banach space having property KM . Thenevery bounded sequence in X ∗ admits a µ ( X ∗ , X ) -convergent convex blocksubsequence. In general, property KM w does not imply property (K). For instance, itis easy to see that c fails property (K) (see, e.g., [5, p. 4998]). A result of Ørno and Valdivia (see, e.g., [15, Theorem 5.53]) states thatevery µ ( X ∗ , X )-convergent sequence in X ∗ is norm convergent whenever theBanach space X contains no subspace isomorphic to ℓ (and conversely).We finish the paper with an application of that result which, in particular,shows that property (K) can be defined via uniform convergence on weaklyprecompact sets. Proposition 3.34.
Let X be a Banach space and ( x ∗ n ) n ∈ N be a sequencein X ∗ which converges to some x ∗ ∈ X ∗ with respect to µ ( X ∗ , X ) . Then ( x ∗ n ) n ∈ N converges to x ∗ uniformly on each weakly precompact subset of X .Proof. Of course, we can assume that x ∗ = 0. Since the absolutely convexhull of a weakly precompact set is weakly precompact (see Subsection 2.1),it suffices to check that ( x ∗ n ) n ∈ N converges to 0 uniformly on each absolutelyconvex weakly precompact set M ⊆ X . To this end, we apply the Davis-Figiel-Johnson-Pe lcz´ynski factorization method to M (see, e.g., [14, p. 250,Lemma 8]) to get a Banach space Y and an operator T : Y → X suchthat M ⊆ T ( B Y ). Since M is weakly precompact, Y contains no subspaceisomorphic to ℓ (see, e.g., [32, Corollary 1.5]). Since T ∗ : X ∗ → Y ∗ is µ ( X ∗ , X )-to- µ ( Y ∗ , Y ) continuous, the sequence ( T ∗ ( x ∗ n )) n ∈ N converges to 0with respect to µ ( Y ∗ , Y ). By the aforementioned result of Ørno and Val-divia, k T ∗ ( x ∗ n ) k Y ∗ → n → ∞ . Sincesup x ∈ M | x ∗ n ( x ) | ≤ sup y ∈ B Y (cid:12)(cid:12) x ∗ n ( T ( y )) (cid:12)(cid:12) = k T ∗ ( x ∗ n ) k Y ∗ for all n ∈ N , we conclude that ( x ∗ n ) n ∈ N converges to 0 uniformly on M . (cid:3) Acknowledgements.
The author thanks A. Avil´es and V. Kadets for valu-able comments on some parts of Subsection 3.3. The research is partiallysupported by
Agencia Estatal de Investigaci´on [MTM2017-86182-P, grantcofunded by ERDF, EU] and
Fundaci´on S´eneca [20797/PI/18].
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