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Problem 6D.18 from Axler's Measure, Integration and Real Analysis is as follows.

Suppose $V$ is a normed vector space such that the dual space $V'$ is a separable Banach space. Prove that V is separable.

I found a proof here that $V$ is separable if $V'$ is separable (and not necessarily Banach). And now I'm slightly confused about why Axler would add the Banach requirement for $V'$. Can you think of any reason for this? Or I misunderstood something?

Jaimi
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    The dual of a normed space is necessarily Banach, see https://math.stackexchange.com/q/3045155/1104384. – Bruno B Feb 27 '25 at 13:22
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    The dual space $V'$ of every normed vector space $V$ is a Banach space (see 6.47 in Measure, Integration & Real Analysis). In the exercise discussed above, I wrote "separable Banach space" instead of "separable normed vector space" because $V'$ is automatically complete and I preferred the shorter phrase. – Sheldon Axler Feb 28 '25 at 03:11

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