fa.functional analysis - Algebraic Dual / Continuous Dual

Ok Ady, since you like CH I will work with CH, and to make your life
easier, I will work with GCH.

Since I do not expect that everybody in MO is aware of various
Banach space constructions, let me give some information on James
tree spaces which are relevant to the question.

A tree is a partially order set $(T,<)$ such that for every $t$ in $T$ the
initial segment ${sin T: s < t}$ is well-ordered under $ < $.
A segment of $T$ is a subset $S$ of $T$ which is:

1. linearly ordered under $ < $ and


2. for all $s, t, win T$ if $s < t < w$ and $s, w in S$ then $tin S$.

The completion of $T$, usually denoted by $c(T)$, is the collection of all initial
segments of $T$ ordered by inclusion. Notice that $c(T)$ contains $T$ and
is much larger than $T$. For instance, if $T$ is the tree of all finite sequences
of natural numbers (usually called the Baire tree, which is clearly countable),
then its completion is the Baire-tree together with its branches (i.e. the
Baire space) and so it has the cardinality of the continuum.

For every tree $T$ the corresponding James tree space $JT$ is defined to
be the completion of $c_{00}(T)$ with the norm:
$$|v| = sup{ (sum_{i=1}^d (sum_{tin S_i} v(t) )^2 )^{1/2} }$$
where the above supremum is taken over all finite families $(S_i)_{i=1}^d$ of
pairwise disjoint segments of $T$. Basic facts (I can provide appropriate
references to anyone who is interested):

• For every tree $T$ the space $JT$ is hereditarily $ell_2$; that is,
  every infinite-dimensional subspace of $JT$ contains a copy of $ell_2$.


• For every tree $T$ the second dual of $JT$ is linearly isometric to
  the James tree space of the completion $c(T)$ of $T$. In particular,
  neither $JT^* $ nor $JT^{**}$ contain a copy of $ell_1$.

Now we come to the specifics of the construction. Remember that we work
with GCH. This implies, in particular, the following: if $X$ is a Banach
space of cardinality $kappa$, then the algebraic dual of $X$ has cardinality
$kappa^+$.

Let $T$ be the tree of all countable subsets of $omega_1$ equipped
with the partial order of end-extension. We have GCH, hence, the tree
is just all sequences of real numbers, and so, it has cardinality
$aleph_1$. The cardinality of the corresponding James tree space is
also $aleph_1$.

The completion $c(T)$ of our tree $T$ is the set of all subsets of
$omega_1$. Hence it has cardinality $2^{aleph_1}$ which is,
under GCH, $aleph_2$. It follows that the cardinality of $JT^{**}$
is $aleph_2$.

Now consider cases.

Case 1: the topological dual $JT^* $ of $JT$ has cardinality strictly
bigger than $aleph_1$. Then we are done: our counterexample is $JT$.

Case 2: the topological dual $JT^* $ of $JT$ has cardinality $aleph_1$.
We are also done: our counterexample is $JT^* $.