Let $X$ be a Banach space, $X' = \mathcal{L}(X, \mathbb{K})$ its dual space. Denote by $\mathcal{B}(X)$ the $\sigma$-algebra of Borel sets and denote by $\sigma(X')$ the $\sigma$-algebra which is generated by all sets of the form $u^{-1}(C)$ for $u \in X'$ and $C \in \mathcal{B}(\mathbb{K})$.

For $X$ separable we have that

$\mathcal{B}(X) = \sigma(X')$ (*)

see e.g. "Gaussian measures in Banach spaces" by Hui-Hsiung Kuo, p. 74 - 75.

Now the author of this book does not bother to discuss the case of $X$ non-separable.

In [1] is a halfway believable counterexample for $X = \ell^2(\mathbb{R})$.

I'm specifically interested in the case $X = \ell^{\infty}$. Does (*) hold in this case and why or why not?

Thanks.

[1] http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_analyst;task=show_msg;msg=1533.0001.0001.0001