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In this answer it is implicitly proved that if $X, Y$ are Banach spaces such that $Y$ embeds continuously and densely into $X$, then $X'$ is a separating subspace of $Y'$.

However, I don't see where density of $Y$ is used in the argument. Am I right to believe that the result is true without density of $Y$?

Guest
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    To interpret $X'$ as a subspace of $Y'$ via the restriction map (the transpose of the inclusion $i:Y\to X$) you need that $i$ has dense range. – Jochen Sep 21 '24 at 11:46
  • @Jochen, Thanks if you add it as an answer, I can accept it. – Guest Sep 27 '24 at 10:36

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