# Is the result still true if we drop completeness? [duplicate]

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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_FarinOct 21 '13 at 7:12

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• What is your argument for the Banach space case? It is important to identify where you use the completeness property. – Christopher A. Wong Feb 27 '13 at 22:22
• 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. – adtx11 Feb 27 '13 at 23:15
• 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 $X$. – Davide Giraudo Mar 2 '13 at 10:58