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Let $X$ be a normed space, $X'$ be its continuous dual, and $X_w'$ be the weak-star version. Apparently, the dual of $X_w'$ is $X$. So the second dual of $X_w'$ should be $X'$. Since $X_w'$ and $X'$ have the same points, the canonical embedding from $X_w'$ to its second dual is onto, meaning that $X_w'$ is homeomorphic to $X'$.

Of course, $X_w'$ isn't the same as $X'$ in general, so I know the above argument is flawed. Can someone please point out my mistake(s)?

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  • the map $i:X_w^* \to X^*$ is not continuous. – Matematleta Mar 04 '17 at 16:10
  • Oh, thank you. I guess a canonical embedding isn't necessarily continuous unless the space being mapped from is normed. So maybe this observation in conjunction with my argument above implies the following: if there exists a norm that generates $X_w'$, then it must be equal to $X'$. – student45 Mar 04 '17 at 16:37
  • If the space is infinite dimensional, not all norms are equivalent. A standard example would be the $p_1$ and $p_2$ norms on $L^{p_1}\cap L^{p_2}.$ So, even if you find a norm that generates $X'_w,$ you cannot say that the identity map is continuous. – Matematleta Mar 04 '17 at 16:44
  • I was actually thinking that the canonical embedding would be an isometric isomorphism and therefore continuous. Is that accurate? – student45 Mar 04 '17 at 16:46

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