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I know how to prove the following exercise ( from Folland) :

If $X$, $Y$ are Banach spaces. $T:X\rightarrow Y$ is a linear map such that $f\circ T\in\operatorname{dual}(X)$ whenever $f\in \operatorname{dual}(Y)$, then $T$ is bounded.

Now I want to know whether the result is still true if $X$ and $Y$ are just normed vector spaces instead of Banach spaces.

Thank you so much.


marked as duplicate by Norbert, Dennis Gulko, user61527, Dominic Michaelis, Lord_Farin Oct 21 '13 at 7:12

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  • $\begingroup$ What is your argument for the Banach space case? It is important to identify where you use the completeness property. $\endgroup$ – Christopher A. Wong Feb 27 '13 at 22:22
  • $\begingroup$ From the given condition I proved that T is a closed linear operator , then by the closed graph theorem it follows that T is bounded. Now to use that theorem we need completeness of both X AND Y. $\endgroup$ – adtx11 Feb 27 '13 at 23:15
  • $\begingroup$ It seems that with Baire's theorem, the result still holds when only $X$ is assumed to be a Banach space. So the question remains when we drop completeness of $X$. $\endgroup$ – Davide Giraudo Mar 2 '13 at 10:58