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Reading through Peter Lax' Functional Analysis book I ran into the following example in section 5:

Let $D$ be some domain in $\mathbb{R}^n$, let $X$ be the space of continuous functions $f$ on $D$ with compact support, and let the norm given by $$||f||_p=\left(\int_D|f(x)|^p\,dx\right)^{1/p},\quad1\leq p.$$ This space is not complete; its completion is denoted by $L^p$.

I have managed showing that the given norm is indeed a norm and that the space is complete. I've learned about completions and that every normed space can be completed. The way this was presented to us is the following:

Completion of a normed space. Let $X$ be a normed space, take $c$ to be the space of Cauchy sequence in $X$ and $c_0\subseteq c$ the space of sequences that convergence to zero. Consider the space $$Z:=c/c_0=\{[x]:x\in c\},$$where $[x]=\{y\in c:y-x\in c_0\}$. Then $Z$ is the completion of $X$, and this completion is unique up to isometries.

Now I'd like to use this, or something similar, to show that the completion of the $X$ of the example indeed equals $L^p$, but I'm not sure how to do this. I've seen some proofs here, but they used some advanced measure theory (Radon measure, Lusin's theorem), or the Hahn Banach Theorem, Riesz' Representation Theorem. I would like a proof coming more from the construction of a completion of a normed space written above (if possible). Any tips are welcome, or if it's not possible, where it goes wrong.

Edit: I've now realized we probably want to show that there exists an isometry between $Z$ from the completion of a normed space part and $L^p$ of the example.

Roberto Rastapopoulos
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  • Do you know about mollifiers? – Mr. Gandalf Sauron Jul 25 '24 at 07:46
  • @Mr.GandalfSauron No, I don't and they are also not mentioned in the book. I believe Brezis' book uses this approach. – Roberto Rastapopoulos Jul 25 '24 at 07:47
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    Basically, given $f$ such that $\int_{D}|f|^{p}<\infty$, you can construct a sequence $f_{n}$ of continuously differentiable compactly supported functions such that $||f_{n}-f||_{L^{p}}\to 0$. Now you have to use the fact that $L^{p}$ is indeed a complete normed space. A standard proof of this fact can be found in Royden's Real Analysis. But you will need some elementary measure theory to prove that a seuqence of functions which converges in measure has a subsequence which converges almost everywhere. – Mr. Gandalf Sauron Jul 25 '24 at 07:51
  • @Mr.GandalfSauron I do know that result in measure theory, I'll give this approach a look, thanks! – Roberto Rastapopoulos Jul 25 '24 at 07:52
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    You can see this approach here – Mr. Gandalf Sauron Jul 25 '24 at 07:54
  • That approach is for infinitely differentiable functions, but I think I should be able to do something similar for only continuous functions. Will first try it myself later and if I get stuck I will use the link you shared. Thanks, again. – Roberto Rastapopoulos Jul 25 '24 at 08:00

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