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Let $(X,\|.\|_{X})$ be a separable Banach space and the associated dual space is denoted by $X^*$. By $w^*$ we shall indicate the weak$-*$ topology on $X^*$.

Let $B_{X^∗}= \{x^∗ \in X^∗ : \|x^∗\|_{X^∗}\leq 1\}$. Since $X$ is separable, the set $B_{X^∗}$ furnished with the relative $w^∗-$topology is compact (by the Alaoglu theorem) and metrizable (see Theorem I.5.85). Note that $$ X^*=\bigcup_{n}{nB_{X^*}} $$ hence $X^∗_{w^∗}$ (the space $X^∗$ furnished with the $w^∗-$topology) is separable.

Can we say that : $X^∗$ is separable in all topologies $\tau$ such that $(X^∗,\tau)^∗ =X$?

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Yes. Separability of a locally convex space is equivalent to the existence of a dense countable-dimensional subspace. All compatible topologies for a dual pair (i.e. those that give the same continuous linear functionals) agree on the closures of convex sets.

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  • $\begingroup$ can you give me a book containing this result? please $\endgroup$ – kaka Hae May 28 at 20:29
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    $\begingroup$ I had to search for one, but according to Google Books (I don't own the book), it appears in section 2.5 of Barrelled Locally Convex Spaces by Carreras and Bonet. $\endgroup$ – Cameron Zwarich May 28 at 20:32
  • $\begingroup$ Thank you very much, @Cameron, for your efforts. $\endgroup$ – kaka Hae May 28 at 21:55

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