SEQUENTIAL PROPERTIES OF THE IN A .

OLAV NYGAARD, AGDER UNIVERSITY COLLEGE

Abstract. We will prove the Eberlein-Smulian theorem. Along with this we will speak about weak convergence of bounded sequences. Finally we will introduce the concept of ”(I)-generating set”, ”Rainwater set” and ”James boundary” and tell some story about these kind of sets, in particular we indi- cate a proof of James’ characterization of reflexive spaces for weakly compactly generated spaces.

1. Four types of compactness and some notation Let A be a subset of a topological space (X, τ). Then • A is compact (C) if every open cover has a finite subcover. • A is countably compact (CC) if every countable cover has a finite subcover. • A is sequentially limit point compact (SLPC) if every sequence (an) has a limit point x, that is, every neighborhood of x contains some ak. • A is sequentially compact (SC) if every sequence in A has a convergent subsequence. Then, in general (C) ⇒ (CC) ⇒ (SLPC) and (SC) ⇒ (SLPC). Recall that (C) is equivalent to the property that every net in A has a convergent subnet. The (CC) and (SLPC) are linked together much the same way, but now only that sequences have convergent subnets. The reason why (SC) is stronger than (SLPC) is that now the convergent subnet can be taken as a subsequence. For first countable spaces (SC) and (SLPC) are equivalent. There are a lot of different types of compactness and a big and powerful theory exists. A beautiful book is [Dug]. In this notes, let X be a Banach space and let X∗ be its dual. The weak topology on X is the weakest topology on X for which all members of X∗ are still continuous. From general theory we now that it is enough to describe a neighborhood base at the origin and that such a base is given by sets of the type ∗ ∗ ∗ ∗ ∗ ∗ W (0; x1, x2, ..., xn; ) = {x ∈ X : |x1(x)|, |x2(x)|...|xn(x)| < }. General theory of weak topologies can be found in [Ru, Ch.3], and more specific descriptions and properties of the neighborhoods for this particular topology can be found in [D, Ch.II]. We will benefit from the following point of view: Let Γ be some subset of X∗ that separates points on X. Then a locally convex Hausdorff topology can be defined ∗ on X like above using the xi ’s only from Γ. From general theory this topology is nothing but the topology of pointwise convergence on Γ. Considering Γ = X ⊂ X∗∗ gives the so called weak-star topology on X∗; the weak and the weak-star topology are the most famous ones on Banach spaces. 1 2 OLAV NYGAARD, AGDER UNIVERSITY COLLEGE

But there are clearly more: We could for example let the extreme points of BX∗ be Γ. That case is interesting and well studied. In the theory of Banach algebras one often works with Γ as the set of multiplicative functionals. Fundamental theorems that we will need are that BX∗ is compact in the weak- star topology (Alaoglu’s theorem) and that BX is weak-star dense in BX∗∗ (Gold- stine’s theorem). Note further that by definition the formal embedding of X into X∗∗ is weak to weak-star continuous. Finally, recall Mazur’s theorem, that for convex sets the -closure and the weak closure coincide.

2. Eberlein’s theorem We are going to prove that if A is a subset of X that is relative sequentially compact in the weak topology, then this set is relatively compact in that topology. For this proof we make some preliminary observations. Observation 2.1. If A ⊂ X is relatively sequentially compact, then A is bounded.

Proof. Suppose A was unbounded. Then we can find a sequence (xn) ⊂ A such that kxnk > n. Thus any subsequence must be unbounded. But according to the Banach-Steinhaus theorem weakly convergent sequences are bounded and hence (xn) can’t have any convergent subsequence.  Observation 2.2. A bounded set A ⊂ X is relatively weakly compact if and only if the weak-star closure of A up in X∗∗ is in X. Proof. Suppose A is relatively weakly compact and look at the weak-star closure A˜ of A in X∗∗. This must be the image of the natural embedding since this embedding is weak to weak-star continuous. Now suppose A˜ ⊂ X. By Alaoglu’s theorem, A˜ is weak-star compact. But on A˜ the weak topology of X and the weak-star topology of X∗∗ coincide by definition.  Observation 2.3. If E is a finite-dimensional linear subspace of X∗, then there 0 ∗ exists a finite set in E ⊂ SX such that for every x ∈ E kx∗k ≤ max |x∗(x)| 2 x∈E0

Proof. Since E is finite-dimensional SE is norm-compact. Thus we can find a 1/4- ∗ ∗ ∗ net F = {x1, x2, ..., xn} for E. And we can choose x1, x2, ....xn in SX such that ∗ ∗ kxi (xi)k > 3/4. Now, let x be arbitrary in E. Then 3 1 1 x∗(x ) = x∗(x ) + (x∗ − x∗)(x ) > − = k k k k k 4 4 2 0 for some appropriate k. Thus, we may simply take E = {x1, x2, ....xn}.  In the last observation here, let us say that ”E0 finds E”. The point is that at least half the norm of E’s members is attained on E0 when E0 finds E. Clearly, for every δ > 0, we can take 1 − δ instead of 2, but that doesn’t help us here. We are ready for Whitley’s proof of Eberlein’s theorem, first published in Math- ematische Annalen in 1967. Eberlein’s original proof was published in 1947. A few years later, Grothendieck showed that Eberlein’s theorem is true in a broader class of locally convex spaces, namely those who are quasi-complete in its Mackey topology. This was done in 1953. Brace gave a proof in 1955 which is the one found SEQUENTIAL PROPERTIES OF THE WEAK TOPOLOGY IN A BANACH SPACE. 3 in the book of Dunford and Schwartz. Pelczynski gave a completely different proof using basic sequences in 1964. We will follow Whitley’s proof more or less as given in [D]. Theorem 2.4. (Eberlein 1947) Let a subset A of a Banach space X be relatively weakly sequentially compact. Then it is relatively weakly compact. Proof. According to Observation 1, A is bounded. By Observation 2, what we then have to do, is to show that every member of the weak-star closure A˜ of A up in X∗∗ are in X. This will be done the following way: For x∗∗ ∈ A˜ we will construct a clever sequence in A. By the relative weak sequential compactness of A, this sequence has a subsequence that converges to some x ∈ X. The cleverness of the sequence is that this subsequence also converges weak-star to that x∗∗. Let x∗∗ ∈ A˜.

Step 1: Constructing the sequence.

∗ Take some x1 ⊂ SX∗ . Look at ∗∗ ∗∗ ∗∗ ∗∗ ∗ W1 = {y ∈ X : |(y − x )(x1)| < 1}. ∗∗ Then clearly W1 is a weak-star neighborhood of x and thus has to contain some a1 from A. Of course ∗∗ ∗ |(x − a1)(x1)| < 1. That was the first element in the sequence. For the next, let ∗∗ ∗∗ ∗∗ ∗∗ E1 = [x , x − a1] (norm-closed of x and x − a1) and choose a finite 0 ∗ ∗ ∗ E1 = {x2, x3, ..., xn(2)} ⊂ SX∗ 0 such that E1 finds E1. Put  1  W = y∗∗ ∈ X∗∗ : |(y∗∗ − x∗∗)(x∗)| < , 1 ≤ i ≤ n(2) . 2 i 2

Inside W2 somewhere there has to sit some a2 ⊂ A, and of course 1 |(x∗∗ − a )(x∗)| < , 1 ≤ i ≤ n(2). 2 i 2 Let ∗∗ ∗∗ ∗∗ E2 = [x , x − a1, x − a2] and pick 0 ∗ ∗ ∗ E2 = {xn(2)+1, xn(2)+2, ..., xn(3)} ⊂ SX∗ 0 such that E2 finds E2. Put  1  W = y∗∗ ∈ X∗∗ : |(y∗∗ − x∗∗)(x∗)| < , 1 ≤ i ≤ n(3) . 3 i 3

Inside W3 again somewhere there has to be some a3 ⊂ A, and 1 |(x∗∗ − a )(x∗)| < , 1 ≤ i ≤ n(3). 3 i 3 Once more, quickly: We put ∗∗ ∗∗ ∗∗ ∗∗ E3 = [x , x − a1, x − a2, x − a3] 4 OLAV NYGAARD, AGDER UNIVERSITY COLLEGE

0 ∗ ∗ ∗ E3 = {xn(3)+1, xn(3)+2, ...xn(4)  1  W = y∗∗ ∈ X∗∗ : |(y∗∗ − x∗∗)(x∗)| < , 1 ≤ i ≤ n(4) . 4 i 3 a4 ∈ W4 ∩ A and go on with E4 to produce W5 and thus a5 and so on. Step 2: Some consequences of the contruction.

Ok, we have the sequence (ai). What properties does it have? We know, since

A is relatively weakly sequentially compact that it has a subsequence (ank ) with a weak limit x and, by Mazur’s theorem x ∈ [ai]. But then,

∗∗ ∗∗ ∗∗ ∗∗ ∗∗ def x − x ∈ [x , x − a1, x − a2, x − a3] = Y. The way Y is built gives us a possibility to control norms there, we have for any y∗∗ ∈ Y that ∗∗ ky k ∗∗ ∗ ≤ sup |y (xm)|. 2 m ∗ So more than half the norm is found along the sequence xm. But this is also the case if we weak-star close Y , and so ∗∗ ∗∗ ∗ kx − xk ≤ 2 sup |(x − x)(xm)|. m

Step 3: x∗∗ = x.

This follows as soon as we can show that for any  > 0 and any m, |(x∗∗ − ∗ x)(xm)| < /2. So, let m be arbitrary. We know that x is the weak limit of (ank ), ∗ so xm(ank − x) → 0. Thus, we can find a natural number p(m) such that nk > p ∗ assures |xm(ank − x)| < /4. Now, if necessary, increase p such that nk > p implies ∗∗ that also |x − ank | < /4. By the triangle inequality ∗∗ ∗ ∗∗ ∗ ∗ ∗∗ |(x − x)(xm)| ≤ |(x − ank )(xm)| + |xm(x − x)|, and we are done.  A remark is that we have only used very elementary means in a clever way. We needed only the Banach-Steinhaus theorem, Mazur’s theorem and Alaoglu’s theorem from , together with basics from topology. Moreover, Observation 1 was used just to avoid stating that A is bounded. Thus, Eberlein’s theorem goes through for bounded sets in any normed space X. If one thought of proving Eberlein’s theorem with pointwise topologies on Γ along the same lines, one quickly comes into many problems. But if Γ has the property that bounded Γ-convergent sequences are weakly convergent, then Eber- lein’s theorem holds for bounded sets. Definition 2.5. A set Γ ⊂ X∗ is called a Rainwater set if every bounded Γ- convergent sequence is weakly convergent. We obtain the following theorem: Theorem 2.6. (Generalized Eberlein’s theorem) Let Γ be a Rainwater set in X∗ and assume that A is a bounded set in X. Then, if A is relatively sequentially Γ-compact, it is relatively weakly compact. SEQUENTIAL PROPERTIES OF THE WEAK TOPOLOGY IN A BANACH SPACE. 5

We will meet Rainwater sets later on, we just mention now that the extreme points of BX∗ provides a such set.

3. Smulian’s theorem Suppose A ⊂ X is relatively weakly compact. Then it is weakly bounded, hence bounded. Every sequence in A must now have a weak limit point. We will show that this limit point in fact is the weak limit of a sequence in A, and this is the essence of Smulian’s theorem. We follow the very wise leadership of Dunford and Schwartz. Recall that a set Γ ⊂ X∗ is called total if its span is weak-star dense in X∗. Theorem 3.1. (Smulian 1940) Suppose a subset A of a Banach space X is rela- tively weakly compact. Then it is relatively weakly sequentially compact. Proof. First we show that we may assume X is separable. To this end, suppose the theorem has been proved in the separable case and let A ⊂ X be relatively weakly compact. Let (an) be a sequence in A. Let Y = [an]. Then Y is a separable, closed subspace. A ∩ Y is relatively weakly compact in Y . Now pick a subsequence (ank ) that converges weakly to some y ∈ Y . Then, since every x∗ ∈ X∗ is nothing but ∗ ∗ an extension of some y ∈ Y ,(ank ) converges weakly to y in X as well. So, the situation is reduced to studying a relatively weakly compact compact set A in a separable Banach space X. We take a sequence (an) in A, which we know has a limit point x ∈ X and we try to find a subsequence of (an) that could have the possibility to converge weakly to x. So, first of all we need a subsequence that has a tendency of converging weakly to something. For this, suppose H is a ∗ countable subset of SX∗ , say (xn). Look at the collection ∗ ∗ ∗ ∗ x1a1 x1a2 x1a3 x1a4 ··· ∗ ∗ ∗ ∗ x2a1 x2a2 x2a3 x2a4 ··· ∗ ∗ ∗ ∗ x3a1 x3a2 x3a3 x3a4 ··· ......

By the Bolzano-Weierstrass theorem, we can pick subsequences such that every horizontal line consist of convergent scalar sequences. Assume so has been done. Then take the diagonal sequence, and denote the corresponding subsequence of (an) ∗ ∗ by (yn). By construction limm x ym exists for every x ∈ H, so some kind of weak convergence seems to emerge. ∗ ∗ Now we will argue that there is a y0 ∈ X such that limm x ym = x y0 for every ∗ x ∈ H. We know (yn) has a weak limit point y0. This means that every weak neighborhood at y0 contains at least one (yn). In particular all the sets ∗ {x ∈ X : |x (x − y0)| < } ∗ ∗ ∗ contains some ym. Since x ym converges for every x ∈ H, the limit has to be x y0. This can be done for every countable H. Let us now make H ”big” to get closer ∗ to weak convergence of (yn). Let (xn) be dense in SX and pick xn ⊂ SX∗ such that ∗ ∗ ∗ xnxn = 1. Let H = (xn). Then H is total in X . To see this, let x ∈ SX be such ∗ ∗ that xnx = 0 for every n and let x ∈ SX∗ . Let  > 0 and find some xn such that ∗ ∗ kx − xnk < . Then, since kx k = 1, |x x| < . Since  was arbitrary, it follows from the Hahn-Banach separation theorem that H is total. 6 OLAV NYGAARD, AGDER UNIVERSITY COLLEGE

To end the proof we show that the subsequence (yn) we obtain with a total H ∗ ∗ ∗ ∗ does the trick, that is x ym → x y0 for every x ∈ X . Suppose not. Then there ∗ ∗ must be some x0 ∈ X , some δ > 0 and some subsequence (ymk ) of (ym) such that ∗ |x0(ymk − y0)| > δ, k = 1, 2, ... ∗ and, by again invoking the Bolzano-Weierstrass theorem, we may assume limk x0ymk 0 exists. But (ymk ) sits in A and has a weak limit point, say y0. Just as above, we ∗ ∗ 0 ∗ ∗ obtain x ymk → x y0 for every x ∈ H and for x0. But then ∗ 0 ∗ ∗ x y0 = x y0 for every x ∈ H, 0 and since H is total, y0 = y0. ∗ ∗ ∗ So |x0(ymk − y0)| > δ, k = 1, 2, ... and still x0(ymk ) → x y0, a hopeless situation. The theorem is proved. 

4. Weak convergence of bounded sequences and a new type of closed In 1963 (see [R] or [D, p. 155]) the following theorem was published by people from Seattle under the pseudomym J. Rainwater: For a bounded sequence in a Banach space X to converge weakly it is enough that it converges pointwise on the extreme points of the unit ball in the dual, BX∗ . It is of course this result that motivates our term Rainwater set. Later on S. Simons (see [S1]or [S2]) gave a completely different argument to show that Rainwater’s theorem is true with any boundary of the dual unit ball. So what is a boundary? Definition 4.1. Let K be a weak-star compact convex subset of X∗. A subset B ⊂ K is called a boundary (or a James boundary) for K if, for every x ∈ X (which we do know attain max on K), the max over K is attained somewhere on B.

Observation 4.2. The extreme points of BX∗ is a boundary for BX∗ . ∗ ∗ Proof. Let x ∈ X. Let m be the max over BX∗ and put J = {x ∈ BX∗ : x (x) = m}. Then J is a weak-star closed face (extreme set), and hence it contains an by the proof of the Krein-Milman theorem.  We formulate Simon’s generalization of Rainwaters theorem:

Theorem 4.3. (Simons, 1972) Every boundary for BX∗ is a Rainwater set. Definition 4.4. Let K be a weak-star compact convex subset of X∗. A subset B ⊂ K is said to (a) (S)-generate K if K is the norm-closed convex hull of B and K is then said to be the (S)-hull of B. (b) (W)-generate K if K is the weak-star-closed convex hull of B and K is then the (W)-hull of B. (c) (I)-generate K if whenever B is built as a countable union, B = ∪nBn, then K is the (I)-hull of B, that is: ! [ ∗ K = k · k-clco w -clcoBn . i SEQUENTIAL PROPERTIES OF THE WEAK TOPOLOGY IN A BANACH SPACE. 7

In other words, K is the (S)-hull of the union of the (W)-hulls of the Bi’s. In this case K is called the (I)-hull of B. It is not difficult to see that any (S)-generating set is (I)-generating and that any (I)-generating set is (W)-generating. Moreover, an example can be given to show that the (S)-hull in general lies strictly in between the (S)-hull and the (W)-hull. It can also easily be seen that to produce the (I)-hull of B we can take increasing unions. In this case the union is itself convex and we end up with only taking norm-closure of the union of the (W)-hulls. Here is an illustration of how the fact that B (I)-generates BX∗ can be used in proving theorems (compare to Rainwater-Simons’ theorems):

Theorem 4.5. Let X be a normed space and suppose B (I)-generates BX∗ . Then B is a Rainwater set.

Proof. Let (xi) be a bounded sequence in X that converges pointwise on B. We must show that (xn) is weakly convergent. Let M be such that kxik, kxk ≤ M for ∗ all i. Pick an arbitrary x ∈ BX∗ and let  > 0. Define ∗ ∗ Bi = {y ∈ B : ∀j ≥ i, |y (xj − x)| < }. ∗ ∗ ∗ Then, since y (xi) → y (x) for every y ∈ B,(Bi) is an increasing covering of B. ∗ ∗ ∗ ∗ Since B has property (I), there is a y in some w -clcoBN such that kx −y k < . ∗ ∗ ∗ Note that for every y ∈ w -clco BN , j ≥ N implies that |y (xj − x)| ≤ . Now, the triangle inequality show that for j ≥ N ∗ ∗ ∗ ∗ ∗ ∗ ∗ |x (xj − x)| ≤ |x (xj) − y (xj)| + |y (xj) − y (x)| + |y (x) − x (x)| ≤ (1 + 2M), and hence (xi) converges weakly to x.  The property that B (I)-generates K was introduced by Fonf and Lindenstrauss in a recent paper, see [FL]. Here the following theorem is the main observation: Theorem 4.6. (Fonf-Lindenstrauss, 2003) Suppose B is a boundary for K. Then B (I)-generates K. Note that Simons’ theorem follows from the two theorems above. Another corol- lary is the following:

∗ Corollary 4.7. Suppose every x ∈ BX∗ attain its norm on BX . Then X is a Grothedieck space, that is, every weak-star convergent sequence in X∗ is weakly convergent.

∗ Proof. That every x ∈ BX∗ attain its norm on BX tells us exactly that BX is a boundary for BX∗∗ . By the Fonf-Lindenstrauss theorem BX (I)-generates BX∗∗ . ∗ ∗ Now, let (xn) be a weak-star convergent sequence in X . Then, by the Banach- Steinhaus theorem, it is bounded. By definition it converges pointwise on BX . Hence, by Theorem 4.5 it is weakly convergent.  A much stronger theorem than the above corollary is true. This is a difficult theorem of R. C. James and reads as follows: Let X be a Banach space. Suppose ∗ every x ∈ BX∗ attain its norm on BX . Then X is reflexive. James’ theorem for a big class of spaces follows, however from the above corollary and Eberlein’s theorem: 8 OLAV NYGAARD, AGDER UNIVERSITY COLLEGE

Corollary 4.8. Let X be a Banach space such that BX∗ is sequentially weak-star ∗ compact. If every x ∈ BX∗ attain its norm on BX , then X is reflexive. Proof. We need only observe that any with a weak-star sequen- tially compact dual unit ball is reflexive. To prove this we will show that then BX∗ is weakly compact, which is equivalent to X being reflexive. By Eberlein’s theorem we need only show it is weakly sequentially compact and so it clearly is, by the Grothendieck property.  Which spaces have weak-star sequentially compact dual unit balls? Well, sep- arable spaces for sure, since then BX∗ in its weak-star topology is metrizable. It is a theorem (not very difficult, one only needs the so called Davis-Figiel-Johnson- Pelczynski construction) of Amir and Lindenstrauss that every subspace of a weakly compactly generated space (WCG-space) is also of this type. A WCG-space is a space where one can find a weakly compact set such that the span of that set is dense.

Definition 4.9. A Banach space such that BX (I)-generates BX∗∗ is called an (I)-space. Note that by Goldstine’s theorem, every Banach space is a (W)-space (the defi- nition should be obvious) and that the (S)-spaces are exatly the reflexive spaces. Question 4.10. Who are the (I)-spaces?

References D. J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92, Springer, 1984. DU. J. Diestel and J. J. Uhl, jr., Vector Measures, Math. Surveys, vol. 15, Amer. Math. Soc., 1977. Dug. J. Dugundji, Topology, Allyn and Bacon, inc., Boston, 1966. FL. V. P. Fonf and J. Lindenstrauss, Boundaries and generation of convex sets, Israel. J. Math. 136 (2003) 157-172. R. J. Rainwater, Weak convergence of bounded sequences, Proc. Amer. Math. Soc. 14 (1963) 999. Ru. W. Rudin. Functional Analysis. 2. ed. Mc. Graw-Hill (1991). S1. S. Simons A convergence theory with boundary. Pacific J. Math. 40 (1972) 703-708 S2. S. Simons An Eigenvector Proof of Fatou’s Lemma for Continuous Functions. The Math. Int. 17 (No 3) (1995) 67-70