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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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Hahn-Banach has nothing to do with the problem at hand (and one only speaks of functionALs if $Y$ is the ground field). The key words here are uniform continuity and completeness of $Y$. –  t.b. Mar 14 '12 at 1:59
    
@t.b. Thanks. I still need to think about this. I agree that I can not apply that theorem directly. –  user16859 Mar 14 '12 at 4:04
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Yes, a bounded linear operator is Lipschitz continuous by definition: $\|Tx_1 - Tx_2\|_Y \leq \|T\|\,\|x_1 - x_2\|_X$. Lipschitz continuity implies uniform continuity. The reason I phrased it the way I did is that it is a general fact that if $f_0: D \to Y$ is uniformly continuous where $D \subset X$ is dense in a metric space $X$ and $Y$ is a complete metric space then $f_0$ admits a unique extension to a (uniformly) continuous $f: X \to Y$. Applying this in the present situation you get the extension $T$ from this general fact and linearity of $T$ follows from uniqueness of the extension –  t.b. Mar 15 '12 at 2:36
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Oh, no, no Tietze at all. It's exactly the same argument as the one azarel outlines. You'll see that you won't use that $T_0$ is linear when you define $T$ (or $f$ as azarel write), you'll only need that when verifying that it is linear... (and since nobody gave you a vote so far, here we go :)) –  t.b. Mar 15 '12 at 2:50
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That's a consequence of the reverse triangle inequality and the definition of $x_n \to x$: $|\|x_n\| - \|x\|| \leq \|x_n - x\| \to 0$, so $\|x_n\| \to \|x\|$. –  t.b. Mar 15 '12 at 6:17
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up vote 4 down vote accepted

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. –  user16859 Mar 14 '12 at 2:21
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