() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 12, no. 1, 2011 pp. 67-80 Some remarks on stronger versions of the Boundary Problem for Banach spaces Jan-David Hardtke Abstract Let X be a real Banach space. A subset B of the dual unit sphere of X is said to be a boundary for X, if every element of X attains its norm on some functional in B. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of X which is compact in the topology of pointwise convergence on B is already weakly compact. This problem was recently solved by Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others. 2010 MSC: 46A50; 46B50 Keywords: boundary; weak compactness; convex hull; extreme points; ε- weakly relatively compact sets; ε-interchangeable double limits 1. Introduction First we fix some notation: throughout this paper X denotes a real Banach space, X∗ its dual, BX its closed unit ball and SX its unit sphere. For a subset B of X∗ we denote by σB the topology on X of pointwise convergence on B. If A ⊆ X, then co A stands for the convex hull of A and A τ for the closure of A in any topology τ on X, except for the norm closure, which we simply denote by A. Also, we denote by ex C the set of extreme points of a convex subset C of X. Now recall that a subset B of SX∗ is called a boundary for X, if for every x ∈ X there is some b ∈ B such that b(x) = ‖x‖. It easily follows from the Krein-Milman theorem that ex BX∗ is always a boundary for X. In 1980 68 J.-D. Hardtke Bourgain and Talagrand proved in [4] that a bounded subset A of X is weakly compact if it is merely compact in the topology σE , where E = ex BX∗ . In [14] Godefroy asked whether the same statement holds for an arbitrary boundary B, a question which has become known as the Boundary Problem. Long since only partial positive answers were known, for example if X = C(K) for some compact Hausdorff space K (cf. [5, Proposition 3]) or X = ℓ1(I) for some set I (cf. [9, Theorem 4.9]). In [24, Theorem 1.1] the positive answer for L1-preduals is contained. Moreover, the answer is positive if the set A is additionally assumed to be convex (cf. [15, p.44]). It was only in 2008 that the positive answer to the Boundary Problem was found in full generality by Pfitzner in [20]. An important tool in the study of the Boundary Problem is the so called Simons’ equality: Theorem 1.1 (Simons, cf. [23]). If B is a boundary for X, then (1.1) sup x∗∈B lim sup n→∞ x∗(xn) = sup x∗∈BX∗ lim sup n→∞ x∗(xn) holds for every bounded sequence (xn)n∈N in X. In particular, it follows from Theorem 1.1 that the well-known Rainwater’s theorem for the extreme points of the dual unit ball (cf. [21]) holds true for an arbitrary boundary: Corollary 1.2 (Simons, cf. [22] or [23]). If B is a boundary for X, then a bounded sequence (xn)n∈N in X is weakly convergent to x ∈ X iff it is σB - convergent to x. Pfitzner’s proof also uses Simons’ equality, as well as a quantitative version of Rosenthal’s ℓ1-theorem due to Behrends (cf. [3]) and an ingenious variant of Hagler-Johnson’s construction. Next we recall the following known characterization of weak compactness (compare [16, p.145-149], [12, Theorem 5.5 and Exercise 5.19] as well as the proof of [10, Theorem V.6.2]). It is a strengthening of the usual Eberlein- Šmulian theorem. Theorem 1.3. Let A be a bounded subset of X. Then the following assertions are equivalent: (i) A is weakly relatively compact. (ii) For every sequence (xn)n∈N in A we have that ∞ ⋂ k=1 co {xn : n ≥ k} 6= ∅. (iii) For every sequence (xn)n∈N in A there is some x ∈ X such that x∗(x) ≤ lim sup n→∞ x∗(xn) ∀x ∗ ∈ X∗. In [18] Moors proved a statement stronger than the equivalence of (i) and (ii), which also sharpens the result from [4]: Some remarks on stronger versions of the Boundary Problem for Banach spaces 69 Theorem 1.4 (Moors, cf. [18]). A bounded subset A of X is weakly relatively compact iff for every sequence (xn)n∈N in A we have that ∞ ⋂ k=1 coσE {xn : n ≥ k} 6= ∅, where E = ex BX∗ . In particular, A is weakly relatively compact if it is merely relatively countably compact in the topology σE . In fact, Moors gets this theorem as a corollary to the following one: Theorem 1.5 (Moors, cf. [18]). Let A be an infinite bounded subset of X. Then there exists a countably infinite set F ⊆ A with coσE F = coF , where E = ex BX∗ . In particular, for each bounded sequence (xn)n∈N in X there is a subsequence (xnk )k∈N with co σE {xnk : k ∈ N} ⊆ co {xn : n ∈ N}. The object of this paper is to give some results related to Theorem 1.4 in the more general context of boundaries. In particular, we shall see, by a very slight modification of the construction from [20], that a ‘non-relative’ version of 1.4 holds for any boundary B of X, see Theorem 2.15. Since we will also deal with some quantitative versions of Theorem 1.4, it is necessary to introduce a bit more of terminology, which stems from [11]: Given ε ≥ 0, a bounded subset A of X is said to be ε-weakly relatively compact (in short ε-WRC) provided that dist(x∗∗, X) ≤ ε for every element x∗∗ ∈ A w ∗ , where w∗ refers to the weak*-topology of X∗∗. For ε = 0 this is equivalent to the classical case of weak relative compactness. The authors of [11] used this notion to give a quantitative version of the well known theorem of Krein (cf. [11, Theorem 2]). In their proof they made use of double limit techniques in the spirit of Grothendieck. More precisely, they worked with the following definition: Let bounded subsets A of X, M of X∗ and ε ≥ 0 be given. Then A is said to have ε-interchangeable double limits with M if for any two sequences (xn)n∈N in A and (x ∗ m)m∈N in M we have ∣ ∣ ∣ lim n→∞ lim m→∞ x∗m(xn) − lim m→∞ lim n→∞ x∗m(xn) ∣ ∣ ∣ ≤ ε, provided that all the limits involved exist. In this case we write A§ε§M . The connection to ε-WRC sets is given by the following proposition: Proposition 1.6 (Fabian et al., cf. [11]). Let A ⊆ X be bounded and ε ≥ 0. Then the following hold: (i) If A is ε-WRC, then A§2ε§BX∗. (ii) If A§ε§BX∗, then A is ε-WRC. In case ε = 0 this is the classical Grothendieck double limit criterion. For various other quantitative results on weak compactness we refer the interested reader to [2], [6], [7] and [11]. For some related results on weak sequential completeness, see also [17]. 70 J.-D. Hardtke We are now ready to formulate and prove our results. However, it should be added that all of them can easily be derived from already known results and techniques. 2. Results and proofs We begin with a quantitative version of Theorem 1.3. First we prove an easy lemma that generalizes the equivalence of (ii) and (iii) in said theorem (the proof is practically the same). Lemma 2.1. Let B be a subset of BX∗ that separates the points of X and let (xn)n∈N be a sequence in X as well as x ∈ X and ε ≥ 0. Then the following assertions are equivalent: (i) x ∈ ⋂∞ k=1 coσB ({xn : n ≥ k} + εBX ). (ii) x∗(x) ≤ lim supn→∞ x ∗(xn) + ε ∀x ∗ ∈ BX∗ ∩ span B. Proof. First we assume (i). It then directly follows that x∗(x) ∈ co ({x∗(xn) : n ≥ k} + [−ε, ε]) ∀k ∈ N ∀x ∗ ∈ BX∗ ∩ span B. Thus we also have x∗(x) ≤ supn≥k x ∗(xn) + ε for all k ∈ N and all x ∗ ∈ BX∗ ∩ span B and the assertion (ii) follows. Now we assume that (ii) holds and take k ∈ N arbitrary. Suppose that x 6∈ coσB ({xn : n ≥ k} + εBX ) . Then by the separation theorem we could find a functional x∗ ∈ (X, σB ) ′ = span B with ‖x∗‖ = 1 and a number α ∈ R such that x∗(y) ≤ α < x∗(x) ∀y ∈ coσB ({xn : n ≥ k} + εBX ) . It follows that lim sup n→∞ x∗(xn) + ε ≤ α < x ∗(x), a contradiction which ends the proof. � Now we can give a quantitative version of the first equivalence in Theorem 1.3. Theorem 2.2. Let A ⊆ X be bounded and ε ≥ 0. If for each sequence (xn)n∈N in A we have (2.1) ∞ ⋂ k=1 co ({xn : n ≥ k} + εBX ) 6= ∅, then A is 2ε-WRC. If A is ε-WRC, then (2.2) ∞ ⋂ k=1 co ({xn : n ≥ k} + rBX ) 6= ∅ holds for every sequence (xn)n∈N in A and every r > ε. Some remarks on stronger versions of the Boundary Problem for Banach spaces 71 Proof. First we assume that (2.1) holds for every sequence in A. Let (xn)n∈N and (x∗m)m∈N be sequences in A and BX∗ , respectively, such that the limits lim n→∞ lim m→∞ x∗m(xn) and lim m→∞ lim n→∞ x∗m(xn) exist. By assumption, we can pick an element x ∈ ∞ ⋂ k=1 co ({xn : n ≥ k} + εBX ) . From Lemma 2.1 we conclude that (2.3) lim inf n→∞ x∗(xn) − ε ≤ x ∗(x) ≤ lim sup n→∞ x∗(xn) + ε ∀x ∗ ∈ BX∗ . It follows that (2.4) ∣ ∣ ∣ x∗m(x) − lim n→∞ x∗m(xn) ∣ ∣ ∣ ≤ ε ∀m ∈ N. Now take a weak*-cluster point x∗ ∈ BX∗ of the sequence (x ∗ m)m∈N. Then (2.5) lim n→∞ lim m→∞ x∗m(xn) = lim n→∞ x∗(xn). By (2.3) we have (2.6) ∣ ∣ ∣ x∗(x) − lim n→∞ x∗(xn) ∣ ∣ ∣ ≤ ε. Since x∗(x)−limm→∞ limn→∞ x ∗ m(xn) is a cluster point of the sequence (x ∗ m(x)− limn→∞ x ∗ m(xn))m∈N it follows from (2.4) that (2.7) ∣ ∣ ∣ x∗(x) − lim m→∞ lim n→∞ x∗m(xn) ∣ ∣ ∣ ≤ ε. From (2.5), (2.6) and (2.7) we get ∣ ∣ ∣ lim m→∞ lim n→∞ x∗m(xn) − lim n→∞ lim m→∞ x∗m(xn) ∣ ∣ ∣ ≤ 2ε. Thus we have proved A§2ε§BX∗ . Hence, by Proposition 1.6, A is 2ε-WRC. Now assume that A is ε-WRC and take any sequence (xn)n∈N in A as well as r > ε. Let x∗∗ ∈ A w ∗ be a weak*-cluster point of (xn)n∈N. Since A is ε-WRC there is some x ∈ X such that ‖x∗∗ − x‖ ≤ r. For every x∗ ∈ BX∗ the number x ∗∗(x∗) is a cluster point of the sequence (x∗(xn))n∈N and thus x∗(x) ≤ ‖x − x∗∗‖ ‖x∗‖ + x∗∗(x∗) ≤ r + lim sup n→∞ x∗(xn). Lemma 2.1 now yields x ∈ ∞ ⋂ k=1 co ({xn : n ≥ k} + rBX ) and the proof is finished. � As an immediate corollary we get 72 J.-D. Hardtke Corollary 2.3. If A ⊆ X is bounded and ε ≥ 0 such that ∞ ⋂ k=1 (co {xn : n ≥ k} + εBX ) 6= ∅ for every sequence (xn)n∈N in A, then A is 2ε-WRC. Now we can also prove a quantitative version of Theorem 1.4: Corollary 2.4. Let A ⊆ X be bounded, ε ≥ 0 and E = ex BX∗ . If for each sequence (xn)n∈N in A we have that ∞ ⋂ k=1 (coσE {xn : n ≥ k} + εBX ) 6= ∅, then A is 2ε-WRC. Proof. Let (xn)n∈N be a sequence in A. By means of Theorem 1.5 and an easy diagonal argument we can find a subsequence (xnk )k∈N such that co σE {xnk : k ≥ l} ⊆ co {xn : n ≥ l} for all l (compare [18, Corollary 0.2]). It then follows from our assumption that ∞ ⋂ l=1 (co {xn : n ≥ l} + εBX ) 6= ∅. Hence, by Corollary 2.3, A is 2ε-WRC. � Next we observe that Moors’ Theorem 1.5 does not only work for the extreme points of BX∗ but also for any weak*-separable boundary. Theorem 2.5. Let B be a weak*-separable boundary for X and A a bounded infinite subset of X. Then there is a countably infinite set F ⊆ A such that coF = coσB F . In particular, for every bounded sequence (xn)n∈N in X there exists a subsequence (xnk )k∈N with co σB {xnk : k ∈ N} ⊆ co {xn : n ∈ N}. Proof. The proof is completely analogous to that of Theorem 1.5 given in [18], in fact it is even simpler, so we shall only sketch it. Arguing by contradiction, we suppose that for each countably infinite subset F of A there is an element z ∈ coσB F \ coF . Then we can show exactly as in [18] (using the Bishop-Phelps theorem (cf. [13, Theorem 5.5]) and the Hahn-Banach separation theorem) that for every sequence (xn)n∈N in A for which the set {xn : n ∈ N} is infinite, there is an element (2.8) x ∈ ∞ ⋂ k=1 coσB {xn : n ≥ k} \ co {xn : n ∈ N} . We remark that the weak*-separability of B is not needed for this step. Next we fix a sequence (xn)n∈N in A whose members are distinct and a countable weak*-dense subset {x∗m : m ∈ N} of B. By the usual diagonal ar- gument we may select a subsequence (again denoted by (xn)n∈N) such that limn→∞ x ∗ m(xn) exists for all m. Some remarks on stronger versions of the Boundary Problem for Banach spaces 73 We then choose an element x according to (2.8) and conclude that for each m ∈ N we have limn→∞ x ∗ m(xn) = x ∗ m(x). Now let x∗ ∈ B be arbitrary. Again as in [18] we will show that limn→∞ x ∗(xn) = x∗(x). Suppose that this is not the case. Then there is an ε > 0 such that |x∗(x) − x∗(xn)| > ε for infinitely many n ∈ N. Let us assume x ∗(xn) > ε + x∗(x) for infinitely many n and arrange these indices in an increasing se- quence (nk)k∈N. By (2.8) we can find z ∈ ∞ ⋂ l=1 coσB {xnk : k ≥ l} \ co {xnk : k ∈ N} . It follows that x∗m(z) = limk→∞ x ∗ m(xnk ) = x ∗ m(x) for all m and since {x ∗ m : m ∈ N} is weak*-dense in B this implies x∗(x) = x∗(z), whereas on the other hand x∗(z) ≥ ε + x∗(x), a contradiction. Thus (xn)n∈N is σB -convergent to x and hence, by Corollary 1.2 it is also weakly convergent to x, which in turn implies x ∈ co {xn : n ∈ N}, contradict- ing the choice of x. � Note that the assumption of weak*-separability of B is fulfilled, in particular, if X is separable, for then the weak*-topology on BX∗ is metrizable. As an immediate corollary we get 2.4 for weak*-separable boundaries. Corollary 2.6. Let B be a boundary for X and A a bounded subset of X as well as ε ≥ 0. If B is weak*-separable (in particular, if X is separable) and for each sequence (xn)n∈N in A we have ∞ ⋂ k=1 (coσB {xn : n ≥ k} + εBX ) 6= ∅, then A is 2ε-WRC. Proof. Exactly as the proof of Corollary 2.4. � Let us now consider Banach spaces of a certain type, namely the case X = C(K) for some compact Hausdorff space K or X = ℓ1(I) for some in- dex set I. In [5] respectively [9] Cascales et al. found the positive solution to the Boundary Problem for these types of spaces. In fact, they even proved a stronger statement, namely that in the above cases the space (X, σB) is an- gelic1 for every boundary B of X. In order to get the statement of Corollary 2.6 for arbitrary boundaries in C(K)- and ℓ1(I)-spaces we shall need the following easy lemma. Lemma 2.7. Let T and S be subsets of X∗ such that for every countable set D ⊆ X and every x∗ ∈ T there is some y∗ ∈ S such that x∗(x) = y∗(x) for all x ∈ D. Then for every countable set D ⊆ X we have coσS D ⊆ coσT D. 1See [5] or [12] for the definition and background. 74 J.-D. Hardtke Proof. Let D ⊆ X be countable and take any x ∈ coσS D. Further, fix x∗1, . . . , x ∗ n ∈ T and ε > 0. By assumption we can find y ∗ 1 , . . . , y ∗ n ∈ S such that x∗i (y) = y ∗ i (y) ∀y ∈ D ∪ {x} ∀i = 1, . . . , n. But then the same equality holds for every y ∈ co (D ∪ {x}) and since x ∈ coσS D we may select some y ∈ co D with |y∗i (x) − y ∗ i (y)| ≤ ε for all i = 1, . . . , n. It follows that |x∗i (x) − x ∗ i (y)| ≤ ε for i = 1, . . . , n and the proof is finished. � According to the results of Cascales et al. the condition of Lemma 2.7 is fulfilled if X = C(K) or X = ℓ1(I), T = ex BX∗ and S is any boundary for X (see [5, Lemma 1] for X = C(K) and the proof of [9, Theorem 4.9] for X = ℓ1(I)), thus we immediately get the following lemma. Lemma 2.8. If X = C(K) for some compact Hausdorff space K or X = ℓ1(I) for some index set I and B is any boundary for X, then for every countable set D ⊆ X we have coσB D ⊆ coσE D, where E = ex BX∗ . From Lemma 2.8 and Corollary 2.4 we now get the desired result. Corollary 2.9. If X = C(K) for some compact Hausdorff space K or X = ℓ1(I) for some set I and B is any boundary for X as well as A ⊆ X a bounded set and ε ≥ 0 such that for every sequence (xn)n∈N in A we have ∞ ⋂ k=1 (coσB {xn : n ≥ k} + εBX ) 6= ∅, then A is 2ε-WRC. Next we turn to spaces not containing isomorphic copies of ℓ1. It is known that for such spaces one has coγ B = BX∗ for every boundary B of X, where we denote by γ the topology on X∗ of uniform convergence on bounded countable subsets of X (cf. [8, Theorem 5.4]). We will also need two easy lemmas. Lemma 2.10. Let A ⊆ X and S ⊆ X∗ be bounded as well as ε ≥ 0 such that A§ε§S. Then we also have A§ε§S γ . Proof. Let (xn)n∈N and (x ∗ m)m∈N be sequences in A and S γ , respectively, such that the limits lim n→∞ lim m→∞ x∗m(xn) and lim m→∞ lim n→∞ x∗m(xn) exist. For each m ∈ N we can pick a functional x̃∗m ∈ S with |x∗m(xn) − x̃ ∗ m(xn)| ≤ 1 m ∀n ∈ N. By the usual diagonal argument, choose a subsequence (xnk )k∈N such that limk→∞ x̃ ∗ m(xnk ) exists for all m. It then easily follows that lim m→∞ lim n→∞ x∗m(xn) = lim m→∞ lim k→∞ x̃∗m(xnk ) and lim n→∞ lim m→∞ x∗m(xn) = lim k→∞ lim m→∞ x̃∗m(xnk ). Some remarks on stronger versions of the Boundary Problem for Banach spaces 75 Since A§ε§S, we conclude that ∣ ∣ ∣ lim n→∞ lim m→∞ x∗m(xn) − lim m→∞ lim n→∞ x∗m(xn) ∣ ∣ ∣ ≤ ε, finishing the proof. � Lemma 2.11. Let (xn)n∈N be a bounded sequence in X and B ⊆ BX∗ such coγ B = BX∗ . Then coσB {xn : n ∈ N} = co {xn : n ∈ N} . Proof. Take x ∈ coσB {xn : n ∈ N} and let ε > 0 and x ∗ 1, . . . , x ∗ k ∈ BX∗ be arbitrary. By assumption, we can find x̃∗1, . . . , x̃ ∗ k ∈ co B such that for i = 1, . . . , k we have |x̃∗i (xn) − x ∗ i (xn)| ≤ ε ∀n ∈ N and |x̃ ∗ i (x) − x ∗ i (x)| ≤ ε. It follows that |x̃∗i (y) − x ∗ i (y)| ≤ ε ∀y ∈ co ({xn : n ∈ N} ∪ {x}) ∀i = 1, . . . , k. Now take some element y ∈ co {xn : n ∈ N} with |x̃ ∗ i (y) − x̃ ∗ i (x)| ≤ ε for all i = 1, . . . , k. Employing the triangle inequality we can deduce |x∗i (x) − x ∗ i (y)| ≤ 3ε, which ends the proof. � As an immediate consequence of Lemma 2.11, Corollary 2.3 and the afore- mentioned result [8, Theorem 5.4] we get the following corollary. Corollary 2.12. Suppose ℓ1 6⊆ X and let B be a boundary for X. If A ⊆ X is bounded and ε ≥ 0 such that for each sequence (xn)n∈N in A we have ∞ ⋂ k=1 (coσB {xn : n ≥ k} + εBX ) 6= ∅, then A is 2ε-WRC. We can further get a kind of ‘boundary double limit criterion’. Proposition 2.13. Let B be a boundary for X as well as ε ≥ 0 and A ⊆ X be bounded such that A§ε§B. Then A is 2ε-WRC. If ℓ1 6⊆ X, then A is even ε-WRC. Proof. From [7, Theorem 3.3] it follows that we also have A§ε§ co B. Since B is a boundary for X the Hahn-Banach separation theorem implies BX∗ = cow ∗ B. Therefore it follows from [2, Lemma 3] that A§2ε§BX∗ . Thus by (ii) of Proposition 1.6 A is 2ε-WRC.2 Moreover, if ℓ1 6⊆ X then we even have BX∗ = co γ B by the already cited [8, Theorem 5.4]. Hence A§ε§BX∗ by Lemma 2.10, thus A is ε-WRC. � 2This proof also works under the weaker assumption that B is only norming for X, i.e. ‖x‖ = sup b∈B b(x) for all x ∈ X, because in this case we also have BX∗ = co w ∗ B by the separation theorem. 76 J.-D. Hardtke Our final aim in this note is to prove a ‘non-relative’ version of Theorem 1.4 for arbitrary boundaries. To do so, we will use the techniques of Pfitzner from [20]. More precisely, we can get the following slight generalization of the “in particular case” of [20, Proposition 8]. Recall that an ℓ1-sequence in X is simply a sequence equivalent to the canonical basis of ℓ1. Proposition 2.14. Let B be a boundary for X. If A ⊆ X is bounded and for every sequence (xn)n∈N in A we have (2.9) A ∩ ∞ ⋂ k=1 coσB {xn : n ≥ k} 6= ∅, then A does not contain an ℓ1-sequence. Proof. The proof is completely analogous to that of [20, Proposition 8], there- fore we will only give a very brief sketch. We use the notation and definitions from [20]. Arguing by contradiction, we assume that there is an ℓ1-sequence (xn)n∈N in A. By [20, Lemma 2] we may assume that (xn)n∈N is δ-stable. We take a sequence (αk)k∈N of positive numbers decreasing to zero. By [20, Lemma 7] we can find ε ≥ 1/2δ̃B(xn) = 1/2δ̃(xn) > 0, a sequence (bk)k∈N in B and a tree (Ωσ)σ∈S such that for each k ∈ N and every σ, σ ′ ∈ Sk with σk = 0 and σ′k = 1 we have bk(xn − xn′ ) ≥ 2ε(1 − αk) ∀n ∈ Ωσ, n ′ ∈ Ωσ′ . It follows that the same inequality holds for every x ∈ coσB {xn : n ∈ Ωσ} and x′ ∈ coσB {xn′ : n ′ ∈ Ωσ′}. Now using our hypothesis we can proceed completely analogous to the proof of the claim in [20, Proposition 8] to find a sequence (ym)m∈N in A∩ ⋂∞ k=1 coσB {xn : n ≥ k} such that bk(ym − ym′ ) ≥ 2ε(1 − αk) ∀m ≤ k < m ′. Next we take an element y ∈ A ∩ ∞ ⋂ k=1 coσB {ym : m ≥ k} . As in the proof of [20, Proposition 8] we put x = ∞ ∑ m=1 2−m(ym − y) and proceed again exactly as in the proof of [20, Proposition 8] to show that ‖ym − y‖ ≤ 2ε for all m and ‖x‖ = 2ε. Finally, taking a functional b ∈ B with b(x) = ‖x‖ we obtain b(y) = 2ε + b(y) and with this contradiction the proof is finished. � Now we can get the final result. Some remarks on stronger versions of the Boundary Problem for Banach spaces 77 Theorem 2.15. Let B be a boundary for X and A ⊆ X be bounded. Then the following assertions are equivalent: (i) A is countably compact in the topology σB . (ii) For every sequence (xn)n∈N in A we have A ∩ ∞ ⋂ k=1 coσB {xn : n ≥ k} 6= ∅. (iii) For every sequence (xn)n∈N in A there is some x ∈ A with x∗(x) ≤ lim sup n→∞ x∗(xn) ∀x ∗ ∈ span B. (iv) A is weakly compact. Proof. The implications (i) ⇒ (ii) and (iv) ⇒ (i) are clear and the equivalence of (ii) and (iii) follows from Lemma 2.1. It only remains to prove (ii) ⇒ (iv). Let us assume that (ii) holds and take an arbitrary sequence (xn)n∈N in A. By Proposition 2.14 no subsequence of (xn)n∈N is an ℓ 1-sequence and thus Rosenthal’s theorem (cf. [3] or [1, Theorem 10.2.1]) applies to yield a subse- quence (xnk )k∈N which is weakly Cauchy. Now choose an element x ∈ A ∩ ∞ ⋂ l=1 coσB {xnk : k ≥ l} . It easily follows that limk→∞ b(xnk ) = b(x) for all b ∈ B. By Corollary 1.2 (xnk )k∈N is weakly convergent to x. Thus we have shown that A is weakly sequentially compact. Hence it is also weakly compact, by the Eberlein-Šmulian theorem. � Remark 2.1. It is proved in [11, Remark 10] that for X = ℓ1 the statement BX§ε§BX∗ is false for every 0 < ε < 2. An alternative proof of this fact is given [6, Example 5.2]. It is further proved in [11, Remark 10] that every separable Banach space X which contains an isomorphic copy of ℓ1 can be equivalently renormed such that, in this renorming, the statement BX§ε§BX∗ is false for every 0 < ε < 2. The proof makes use of the notion of octahedral norms. We wish to point out here that the argument from [6, Example 5.2] can be carried over to arbitrary Banach spaces containing a copy of ℓ1, precisely we have the following proposition. Proposition 2.16. If X is a (not necessarily separable) Banach space which contains ℓ1 than the statement BX§ε§BX∗ (in the original norm of X) is false for every 0 < ε < 2. Proof. Take 0 < ε < 2 arbitrary and fix 0 < δ < 1 such that 2(1 − δ) > ε. Since X contains ℓ1 we may find, with the aid of James’ ℓ1-distortion theorem (cf. [1, Theorem 10.3.1]), a sequence (xn)n∈N in the unit sphere of X such that T : ℓ1 → X defined by T y = ∞ ∑ k=1 αkxk ∀y = (αn)n∈N ∈ ℓ 1 78 J.-D. Hardtke is an isomorphism (onto U = ran T ) with ∥ ∥T −1 ∥ ∥ ≤ (1 − δ)−1. Consequently, the adjoint T ∗ : U ∗ → ℓ∞ is as well an isomorphism with ∥ ∥(T ∗)−1 ∥ ∥ ≤ (1−δ)−1. Now we can define as in [6, Example 5.2] for each n ∈ N a norm one functional y∗n ∈ ℓ ∞ by y∗n(m) = { 1, if m ≤ n −1, if m > n. Put u∗n = (T ∗)−1y∗n for all n ∈ N. Then ‖u ∗ n‖ ≤ (1 − δ) −1 and hence by the Hahn-Banach extension theorem we can find x∗n ∈ BX∗ with x ∗ n|U = (1 − δ)u ∗ n for all n ∈ N. It follows that ∣ ∣ ∣ lim n→∞ lim m→∞ x∗n(xm) − lim m→∞ lim n→∞ x∗n(xm) ∣ ∣ ∣ = 2(1 − δ) > ε and the proof is finished. � In the notation of [6] we have proved γ(BX ) = 2 for every Banach space X containing an isomorphic copy of ℓ1, which implies that the value of BX under all other measures of weak non-compactness considered in [6] is equal to one (again compare [6, Example 5.2]). So in a certain sense a Banach space containing ℓ1 is ‘as non-reflexive as possible’. Remark 2.2. Shortly after the first version of this paper was published on the web, the author received a message from Prof. Warren B. Moors, who kindly pointed out to him that the above Lemma 2.8 probably also holds true if X is an L1-predual 3 (which includes all C(K)-spaces), refering to the paper [19]. Indeed, from [19, Theorem 3] one can easily get the following result: if B is a boundary for the L1-predual X and E = ex BX∗ , then coσB {xn : n ∈ N} ⊆ co σE {xn : n ∈ N} holds for every sequence (xn)n∈N in X. For the proof just apply [19, Theorem 3] to the countable set coQ {xn : n ∈ N} consisting of all convex combinations of the xn’s with rational coefficients. It follows that Corollary 2.9 also carries over to arbitrary boundaries of L1-preduals. Acknowledgements. The author wishes to express his gratitude to Prof. Warren B. Moors for providing him with the important hint already mentioned in the remark above and to the anonymous referee for multiple comments and suggestions (in particular for proposing Lemma 2.7) which improved the expo- sition of the results. 3Recall that a Banach space X is called an L1-predual if X ∗ is isometric to L1(µ) for some suitable measure µ. Some remarks on stronger versions of the Boundary Problem for Banach spaces 79 References [1] F. Albiac and N. Kalton, Topics in Banach Space Theory, Springer Graduate Texts in Mathematics vol.233, 2006 [2] C. Angosto and B. Cascales, Measures of weak noncompactness in Banach spaces, Topol- ogy Appl. 156, no. 7 (2009), 1412–1421. [3] E. Behrends, New proofs of Rosenthal’s ℓ1-theorem and the Josefson-Nissenzweig theo- rem, Bull. Polish Acad. Sci. Math. 43 (1996), 283–295. [4] J. Bourgain and M. Talagrand, Compacité extremalé, Proc. Amer. Math. Soc. 80 (1980), 68–70. [5] B. Cascales and G. Godefroy, Angelicity and the boundary problem, Mathematika 45 (1998), 105–112. [6] B. Cascales, O. Kalenda and J. Spurný, A quantitative version of James’ compactness theorem, (http://arxiv.org/abs/1005.5693). [7] B. Cascales, W. Marciszewski and M. Raja, Distance to spaces of continuous functions, Topology Appl. 153, no. 13 (2006), 2303–2319. [8] B. Cascales, M. Muñoz and J. Orihuela, James boundaries and σ-fragmented selectors, Studia Math. 188, no. 2 (2008), 97–122. [9] B. Cascales and R. Shvydkoy, On the Krein-Šmulian Theorem for weaker topologies, Illinois J. Math. 47 (2003), 957–976. [10] N. Dunford and J. Schwartz, Linear Operators Part I: General Theory, Interscience Publishers, New York, 1958 [11] M. Fabian, P. Hájek, V. Montesinos and V. Zizler, A quantitative version of Krein’s theorem, Rev. Mat. Iberoamericana 21, no. 1 (2005), 237–248. [12] K. Floret, Weakly compact sets, Springer Lectures Notes in Mathematics vol. 801, 1980 [13] V. P. Fonf, J. Lindenstrauss and R. R. Phelps, Infinite dimensional convexity, in: Hand- book of the geometry of Banach spaces, vol. 1 (W. B. Johnson and J. Lindenstrauss, eds.), North-Holland, 2001, pp. 599–670. [14] G. Godefroy, Boundaries of a convex set and interpolation sets, Math. Ann. 277 (1987), 173–184. [15] G. Godefroy, Five lectures in Geometry of Banach spaces, Seminar on Functional Ana- lysis (1987), 9–67. [16] R. B. Holmes, Geometric Functional Analysis and its Applications, Springer Graduate Texts in Mathematics vol. 24, 1975. [17] O. F. K. Kalenda, H. Pfitzner and J. Spurný, On quantification of weak sequential completeness, preprint, 2010, (http://www.arxiv.org/abs/1011.6553v1). [18] W. B. Moors, A characterisation of weak compactness in Banach spaces, Bull. Austral. Math. Soc. 55 (1997), 497–501. [19] W. B. Moors and J. Spurný, On the topology of pointwise convergence on the boundaries of L1-preduals, Proc. Amer. Math. Soc. 137 (2009), 1421–1429. [20] H. Pfitzner, Boundaries for Banach spaces determine weak compactness, Invent. Math. 182 (2010), 585–604. [21] J. Rainwater, Weak convergence of bounded sequences, Proc. Amer. Math. Soc. 14 (1963), 999. [22] S. Simons, A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703–708. [23] S. Simons, An eigenvector proof of Fatou’s lemma for continuous functions, Math. Intelligencer 17 (1995), 67–70. [24] J. Spurný, The Boundary Problem for L1-Preduals, Illinois J. Math. 52 (2008), 1183– 1193. (Received November 2010 – Accepted January 2011) 80 J.-D. Hardtke Jan-David Hardtke (hardtke@math.fu-berlin.de) Department of Mathematics, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany Some remarks on stronger versions of the [3pt] Boundary Problem for Banach spaces. By J.-D. Hardtke