Next: The weak-* topology in Z Up: Separable Banach space theory Previous: Banach space preliminaries

# Trees and subspaces of

In this section we prove the following result: For each countable ordinal , there exists a weak-*dense subspace Z of such that . Our proof uses some simple concepts and results concerning trees. We give a self-contained treatment of these auxiliary results.

Definition 3.1   Let denote the set of finite sequences of natural numbers, i.e.,

For , we write

where denotes the length of s, and s(i)=ni for all . In particular is the empty sequence, the unique sequence of length 0. For we denote by the concatenation of s and t, i.e., the sequence of length given by

For , we write to mean that s is an initial segment of t, i.e., and, for all , s(i)=t(i). Given , if let

i.e., s' is the initial segment of s of length . For we put .

Definition 3.2   We define a tree to be a nonempty set which is closed under taking initial segments, i.e., for all , if and then . If T is a tree and , we say that s is a node of T. If s is a node of Tsuch that for all , then s is called an end node; otherwise s is called an interior node of T. Given a tree T, a function is called a path through T if for all we have , where . A tree T is said to be well-founded if it has no path.

Definition 3.3   If T is a tree, let

Note that T' is a subtree of T. We define a transfinite sequence of subtrees of T by

Note that T is well-founded if and only if for some countable ordinal . The least such is called the height of T, denoted h(T). Given a well-founded tree T, we define a function (where denotes the set of countable ordinals) by hT(s)=-1 for and, for , hT(s)=the least such that s is an end node of . In particular . Note that, for all , .

From the definition above, the height of a well-founded tree is a countable ordinal. The following standard theorem shows that the converse holds as well.

Theorem 3.4   For any countable ordinal , we can construct a well-founded tree T such that .

Proof. We prove this by transfinite induction on . For we have . For successor ordinals, note that if , then where . Suppose now that is a limit ordinal, say , where for all . For each n let Tn be a tree of height , and put . Then T is well-founded and, for each , . Thus .

Definition 3.5   Fix an injection with the following properties:
1.
;
2.
implies ;
3.
m<n implies .
Given we refer to as the Gödel number of s. To simplify notation in what follows, we shall often identify sequences with their Gödel numbers, i.e., we write s instead of .

Let , the space of sequences of real numbers converging to 0 indexed by . Then we may identify X* with , the space of absolutely summable sequences of real numbers indexed by . In the rest of this section we shall mainly be interested in the weak-* topology on .

Definition 3.6   For each we define a distinguished point by

Using the convention that sequences are to be identified with their Gödel numbers, we can write ys(t)=t' if , 0otherwise.  Note that, for all , the sequence , converges weak-* to ys in .

Definition 3.7   Given we set

If , let . Note that ZS is a norm closed subspace of . Note also that if and only if .

The next two lemmas imply that, for any well-founded tree T, ZTis weak-* dense in .

Lemma 3.8   If T is a well-founded tree, then for all .

Proof. We proceed by induction on hT(s). If hT(s)=-1 then and . Suppose now that , and that the theorem holds for all t such that . Then for each , , so for all . Since ys is the weak-* limit of the sequence , it follows that , as desired.

Lemma 3.9   If T is a well-founded tree, then ZT is weak-* dense in ; in fact, .

Proof. Let be given. We can write , where is the characteristic function of , i.e., if t=s, 0otherwise. Note that if then , whereas . Also, by the previous lemma, if then , so, if then . Since z is an absolutely summable series, we have

Now, is just a real number since z is absolutely summable, and is the weak-* limit of the sequence , so the first term in this last sum is in ZT(h(T)+1). Likewise , viewed as a series in , converges in norm, and hence converges weak-*; since the functionals and are in ZT(h(T)) for all and all with , it, too, is the weak-* limit of a sequence from ZT(h(T)). Hence z is the sum of two weak-* limits of sequences from ZT(h(T)), whence z is in ZT(h(T)+1) as desired.

Corollary 3.10   If T is a well-founded tree, then .

Thus we have an upper bound on the closure ordinal of ZT in terms of the height of T. To get a lower bound, we use the following technical lemma, which gives us a handle on the growth of the spaces .

Lemma 3.11   Suppose that weak-* in . Let be given and suppose that for all . A sufficient condition for is the existence of such that for all and all .

Proof.

Assume that the stated condition holds. Suppose for a contradiction that , say

Then for all sufficiently large M we have

Fix such an M, with for all and all as well. Then for all sufficiently large k we have

and hence

and hence . Thus is unbounded, contradicting Corollary 2.3.

In particular, we have the following result.

Corollary 3.12   Let be such that, for each , for all but finitely many . Then ZS is weak-* closed.

Proof. By Lemma 3.11 ZS is weak-* sequentially closed. Hence by Corollary 2.11 ZS is weak-* closed.

In order to make use of this lemma, we consider a special class of trees known as smooth trees:

Definition 3.13   For we say s is majorized by t, written , if and for all . For any tree T, we define T* to be the upward closure of T under majorization, i.e.,

A tree T is said to be smooth if it is upward closed under majorization, i.e., T*=T.

Lemma 3.14 (Marcone [15,16])   Let T be a tree. Then T is well-founded if and only if T* is well-founded, in which case h(T)=h(T*).

Proof. Note first that , so if T* is well-founded then so is T. Conversely, suppose T* has a path f; let . Then Tf is a finitely-branching subtree of T, and, since f is a path through T*, Tf must be infinite. Hence by König's Lemma Tf has a path, whence T has a path.

Assuming T and T* are well-founded, we obviously have . For the opposite inequality, we claim that for all s, . (Note that is a finite set, so we may take max rather than sup.) We prove the claim by induction on hT*(s). If hT*(s)=-1 then , so for any t with we have , whence hT(t)=-1 for all such t. Otherwise and we have

This proves our claim. In particular and the proof of the lemma is complete.

Corollary 3.15   For any countable ordinal , there exists a smooth well-founded tree T such that .

Proof. This follows immediately from Theorem 3.4 and the previous lemma.

Lemma 3.16   If T is a smooth tree, then is smooth for all .

Proof. We proceed by induction on . For there's nothing to prove. Assume is smooth, and let be given. Suppose ; since which is smooth, tmust be in ; furthermore, since s is an interior node of , there is an such that . But , so , whence . Finally, smoothness is clearly preserved under intersections, so the induction goes through at limit stages.

Lemma 3.17   Let T be a smooth well-founded tree. Then for all , .

Proof. We proceed by induction on . If there's nothing to prove. Assume and and let be given. Then z is the weak-* limit of some sequence from . Since , we have that any is an interior node of , i.e., and for some M. Since T is smooth, so is , and hence for all . Hence, for each we have and for all . By Lemma 3.11 it follows that . Since s is an arbitrary node in , we have . This shows that . Finally, if is a limit ordinal and for all , it follows easily that for all , whence . This completes the proof.

Corollary 3.18   If T is a smooth well-founded tree, then .

Proof. By Lemma 3.9 we have and . On the other hand, so ; hence, by the previous lemma, so in particular , and hence . This completes the proof.

We now obtain the main result of this section, originally due to McGehee [18]:

Theorem 3.19   For any countable ordinal , there exists a weak-* dense subspace Z of such that .

Proof. Since is a countably infinite set, we may identify with . By Corollary 3.15 let T be a smooth well-founded tree of height . By Lemma 3.9 ZT is weak-* dense in and by Corollary 3.18 we have .

Corollary 3.20   The ordinals which can occur as closure ordinals of subspaces of the dual of a separable Banach space are precisely the countable non-limit ordinals.

Proof. This is immediate from Theorems 2.13 and 3.19. The result is originally due to Sarason [19,20,21] and McGehee [18].

This settles the question of closure ordinals of subspaces of , but what about ? It turns out that the answer is much simpler (see Corollary 3.23 below).

Theorem 3.21   Let X be a Banach space, and let be convex. Then for any , the following conditions are equivalent:
1.
x is in the weak closure of C.
2.
x is in the norm closure of C.
3.
x is the norm limit of a sequence of points in C.
4.
x is the weak limit of a sequence of points in C.

Proof. The equivalence of (1) and (2) follows from the well-known fact (Theorem V.3.13 in [11]) that a convex set is weakly closed if and only if it is norm closed. The implications (2) (3) (4) (1) are all trivial.

Corollary 3.22   If X is a reflexive Banach space and is convex, then the weak-* sequential closure ordinal of C is 0 if C is weak-* closed, and is 1 otherwise.

Proof. If X is reflexive then so is X* (see Corollary II.3.24 in [11]), and hence the weak and weak-* topologies on X*coincide. The result now follows immediately from Theorem 3.21.

We would like to thank Howard Becker for pointing this out to us.

Corollary 3.23   If is convex, then the weak-* (i.e., weak) sequential closure ordinal of C is 0 if C is weak-*(i.e., weakly) closed, 1 otherwise.

We do not know whether the closure ordinal of a convex set in the dual of a separable Banach space can be a limit ordinal.

Next: The weak-* topology in Z Up: Separable Banach space theory Previous: Banach space preliminaries
Stephen G Simpson
1998-10-25