Let $X$ be a Banach space, $X’$ be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about the density characters (which I’ll denote by $d$) and cardinalities of both $X$ and $X’$. Let’s exclude the simplest case that both spaces are norm-separable. And for simplicity let’s assume the generalized continuum hypothesis. Then I can show there are three potential possibilities, two of which I have examples:
- $d(X) = \aleph_0$, $|X| = 2^{\aleph_0}$, $d(X’) = 2^{\aleph_0}$, $|X’| = 2^{\aleph_0}$. This happens, for example, when $X = l^1(\mathbb{N})$ and $X’ = l^\infty(\mathbb{N})$;
- $d(X) = 2^{\aleph_0}$, $|X| = 2^{\aleph_0}$, $d(X’) = 2^{2^{\aleph_0}}$, $|X’| = 2^{2^{\aleph_0}}$. This happens, for example, when $X = l^\infty(\mathbb{N})$ and $X’ = l^\infty(\mathbb{N})’$ or when $X = l^1(2^{\aleph_0})$ and $X’ = l^\infty(2^{\aleph_0})$;
However, I have been failing to find an example for the third case, which is when $d(X) = 2^{\aleph_0}$, $|X| = 2^{\aleph_0}$, $d(X’) = 2^{\aleph_0}$, $|X’| = 2^{\aleph_0}$. So I’ve been wondering if this case is actually possible. Namely, is it possible for $X’$ to have density character $2^{\aleph_0}$, and its unit ball to be weak*-separable, but $X$ is not separable, assuming the generalized continuum hypothesis if that helps?