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Suppose $X,Y$ are two normed spaces and let $X^*,Y^*$ denote their corresponding duals. We are given some bounded linear map $G: Y^* \to X^*$ and asked whether we can we find a corresponging bounded linear map $T:X \to Y$ such that $T^*=G$.

I guess the answer is no. I could not get that far into my argument, but I was thinking about the fact that if $X$ and $Y$ are not finite dimensional, then their duals are strictly bigger. This suggests that the space of bounded linear functions between $X$ and $Y$ is of lower dimension than that of between $X^*$ and $Y^*$. I assume I need some additional argument that this is also true for the set of bounded linear functions between $Y^*$ and $X^*$.

Is there another approach you suggest? If not how can I close this argument? Thanks!

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    Given $G: Y^* -> X^$ dualize it to get $G^ : X^{} -> Y^{}$. Now, $X$ is contained in $X^{}$ and $Y$ in $Y^{}$ so, a natural candidate for $T$ would be $G^*$ restricted to $X$. – Salcio Nov 27 '21 at 15:31
  • But we dont know that the image is in $Y$ right? –  Nov 27 '21 at 16:35

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