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Notation:

All functions here are from $X$ to $\mathbb R$.

$C_c(X)$ = compactly supported continuous functions.

$C_b(X)$ = bounded continuous functions.

$B(X)$ = bounded functions.

$C_0(X)$ = continuous functions that tend to zero (so $X$ has to be locally compact and Hausdorff)


Today I proved that $C_c(\mathbb R)$ is not complete with respect to $\|\cdot\|_\infty$. One can do this by taking $g_n$ to be the function that is zero on $(-\infty,-n]$, linear on $[-n, -n+1]$, $1$ on $[-n+1, n-1]$ and symmetric with respect to the $y$-axis. For $f(x) = e^{-x^2}$ one can show that $\|fg_n - f\|_\infty \to 0$ but $f \notin C_c(\mathbb R)$.

Then I read about completions and wanted to work out the completion of $C_c$(X). (I did this all for $X = \mathbb R$). In any case, my thoughts were as follows: $C_c(X)$ is contained in $B(X)$. But it is a proper subset because the uniform limit of continuous functions is continuous but there are discontinuous bounded functions. The next candidate then seems to be $C_b(X)$ but $f(x) = 1$ is in there and not the uniform limit of $f_n$ in $C_c$. (cannot be because if $f_n$ is zero outside a compact set then $\|f_n - 1\|_\infty = 1$ for all $n$ so this doesn't converge in norm).

The next candidate then is $C_0(X)$ and I'm quite sure that that's the completion of $C_c$ with respect to $\|\cdot\|_\infty$ in $B(X)$. But now I need to show this by showing that $C_0(X)$ is isomorphic to the space of Cauchy sequences in $C_c(X)$ quotient Cauchy sequences that tend to the zero function and I don't really know how to think about this. Can someone please show me how to prove this? Thank you. I want to see this quotient construction and an isomorphism but if there are other ways to show that $C_0$ is the completion of $C_c$ then go ahead and post it, I will upvote it.

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    If $f \in C_0(X)$ find a compact set $K$ such that $|f(x)| \lt \varepsilon$ whenever $x \notin K$. Consider the restriction $f|_K and remember Tietze (again :)).2012-07-11
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    Addendum: I started to think about this because there is an exercise in my notes that asks me to find a Banach space $B$ containing $(C_c, \|\cdot\|_\infty)$ such that the norm on $B$ restricted to $C_c$ is the same as $\|\cdot\|_\infty$. I think $B(X)$ is such a space (easy to show that it's complete), right?2012-07-11
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    Yes, that's right. Note that the completion is unique up to unique isometry, so you only have to find a complete space in which $C_c(X)$ is dense with respect to $\|\cdot\|$ and I suggested how to show that $C_c$ is dense in $C_0$. And yes, we've convinced ourselves at various points over the last half-a-year that $B(X) = \ell^\infty(X)$ is complete...2012-07-11
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    @t.b. I upvoted your comments (as mentioned in my question). But I still want to see the quotient construction with the isomorphism!2012-07-11
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    @t.b. How can I use Tietze here? If I continuously extend $f\mid_K$ to $\mathbb R$, I get a continuous function which is what I had to begin with. Tietze doesn't give me compact support. OTOH, if I take the (usual) $g_n$ (same as I used in my question) then I get a sequence $fg_n$ in $C_c$ converging to $f \in C_0$ in $\|\cdot\|_\infty$.2012-07-11
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    Of course you can use Tietze. Take an open set $U$ with compact closure containing $K$ (you can do that by local compactness of $X$). Define a function $g$ on $K \cup (X \smallsetminus U)$ by $g = f|_K$ on $K$ and $g = 0$ on $X \smallsetminus U$. Observe that $g$ is continuous. Now extend.2012-07-11
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    @t.b. But I could also just use $fg_n$, no?2012-07-11
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    My suggestion works on any locally compact Hausdorff space. But on $\mathbb{R}$, you can use those $g_n$'s, yes.2012-07-11
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    @t.b. Great! Thanks for these comments! Now for maximum awesome you could write an answer that shows the quotient construction with an iso from the quotient thingie to $C_0$ : )2012-07-11
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    Is [this link](http://www.proofwiki.org/wiki/Completion_Theorem_(Metric_Spaces)) good enough? ProofWiki is usually not very good but this entry looks reasonable (I haven't read it).2012-07-11
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    @t.b. It's good. It's what is shown in the notes linked to in the comments to ncmathsadist's answer.2012-07-11
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    @t.b. I was hoping to learn something from seeing it done with the concrete example at hand. For example, next time in the same situation how to see the completion. I can do that if I "see" what the quotient space of sequences looks like.2012-07-11
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    @t.b. But at the moment I can still only *guess* that Cauchy sequences in $C_c$ modulo Cauchy sequences converging to zero (in $\|\cdot\|_\infty$) look like $C_0$. And then tomorrow when I'll replace the sup norm with $\|\cdot\|_p$ I might be stuck (=*head asplode*) again.2012-07-11
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    @t.b. [Or not.](http://chat.stackexchange.com/transcript/message/2918479#2918479) Now I need to come up with an example to which I don't know the answer!2012-07-12

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