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Let $E$ be a dense linear subspace of a normed vector space $X$, and let $Y$ be a Banach space. Suppose $T_{0}\in\mathcal{L}(E,Y)$ is a bounded linear operator from $E$ to $Y$. Show that $T_{0}$ can be extended to $T\in\mathcal{L}(X,Y)$ (by continuity) without increasing its norm.

I have a dumb question: Given the Hahn-Banach theorem, what's to prove here? It seems to be the immediate consequence of that theorem. If I am wrong, please show me how to prove this. Thank you!

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    Finally, problem solved. Thank you so much!2012-03-15

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Hahn-Banach only apply if $Y=\mathbb R$. For this particular problem you want to show that if $(x_n)$ converges to $x$ then $T_0(x_n)$ is a Cauchy sequence and then define $f(x)$ as the limit of the sequence. Finally you need to show that the map is a well-defined bounded linear function.

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    Thanks. I will try to figure out the details.2012-03-14