Assume $ X_1$ and $X_2$ are two Banach Spaces such that $X_1\subset X_2$, i.e., the element belongs to $X_1$ belongs to $X_2$. No assumption on norms. Then I would expect that the dual space of them have the relation such that $$ X_2^\ast\subset X_1^\ast $$ Now, if I know $X_1^\ast = X_2^\ast$, i.e., they share the same element. Then, do I have $X_1=X_2$ as well?
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2Related. You generally have an inclusion $X_2^\ast \subset X_1^\ast$, or rather a canonical injection, only if the inclusion $X_1 \hookrightarrow X_2$ is continuous with dense image. – Daniel Fischer Aug 21 '15 at 13:53
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It might help to contemplate a case where $X\neq X^{**}$. – hardmath Aug 21 '15 at 15:16
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See this http://math.stackexchange.com/questions/966208/are-there-spaces-smaller-than-c-0-whose-dual-is-ell1/967079#967079 – Tomasz Kania Aug 21 '15 at 22:10