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.
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.
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 .
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 .
The next two lemmas imply that, for any well-founded tree T, ZTis weak-* dense in .
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.
be given. We can write
characteristic function of ,
if t=s, 0otherwise. Note that if
Also, by the previous
Since z is an
absolutely summable series, we have
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 .
Assume that the stated condition holds. Suppose for a
In particular, we have the following result.
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:
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,
is a finite set, so we may take max rather
than sup.) We prove the claim by induction on
so for any t with
we have ,
whence hT(t)=-1 for all such t. Otherwise
and we have
Proof. This follows immediately from Theorem 3.4 and the previous lemma.
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.
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.
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 :
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 .
Proof. This is immediate from Theorems 2.13 and 3.19. The result is originally due to Sarason [19,20,21] and McGehee .
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).
Proof. The equivalence of (1) and (2) follows from the well-known fact (Theorem V.3.13 in ) that a convex set is weakly closed if and only if it is norm closed. The implications (2) (3) (4) (1) are all trivial.
Proof. If X is reflexive then so is X* (see Corollary II.3.24 in ), 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.
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.