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I'm studying functional analsyis and I keep wondering if there are some inclusions between the spac eof the continuous function and $L^p$ spaces (it will appear soon a similar question regarding $C_b$ and $C_c$, but let's focus on the supposed simplest case).

So the question is: consider $\Omega$ a bounded open set in $\mathbb{R}^n$, and consider respectively $C^0(\bar{\Omega})$ and $L^p(\Omega)$ (whhere $p\in [1,+\infty]$).

My idea: by the nesting property of $L^p(\Omega)$ ($\Omega$ by construction has finite Lebesgue measure), if $p<q$ then $L^q(\Omega)\subset L^p(\Omega)$. Therefore it is sufficient to prove it for $L^{\infty}(\Omega)$. But now it is quite simple to see that if $f\in C^0(\bar\Omega)$, then $\left\lVert f\right\rVert_{\infty,\bar{\Omega}}=\left\lVert f\right\rVert_{L^{\infty}}$. Thus we have the inclusion $C^0(\bar\Omega)\subset L^{\infty}(\Omega)$, so $C^0(\bar\Omega)\subset L^p(\Omega)$.

Is it true? Am i missing something very simple? I tried to look on literature about these result but maybe it's obvious or totally wrong, we'll see.

Finally, I was also wondering if by this I can easily conclude that $C^k(\bar\Omega)\subset L^p(\Omega)$, as $C^k(\bar\Omega)\subset C^0(\bar\Omega)$ by construction. If the above computations are correct it should immediately follow, but I ask for sure.

Any hint, correction, solution or reference would be much appreciate, thanks in advance.

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  • $\begingroup$ True, but it is simpler to just use the Hölder inequality directly and prove that $$\lVert f\rVert_{L^p(\Omega)}\le C\lVert f\rVert_{C^0}; $$ it is immediate. That's ultimately the same thing you did, actually. $\endgroup$ Feb 5, 2020 at 10:20
  • $\begingroup$ You are not missing anything. These facts are quite trivial and your arguments are correct. $\endgroup$ Feb 5, 2020 at 10:21
  • $\begingroup$ Thanks guys, I thought it was rather simple but I preferred to put things written down also to clarify all the passages :) $\endgroup$
    – qwertyguy
    Feb 5, 2020 at 10:23

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This follows because a continuous function on a closed and bounded set is bounded. So, for any $f \in C(\overline{\Omega})$, we have $$ \biggl(\int_{\Omega} |f(x)|^p dx\biggr)^{1/p} \leq \mathcal{L}^n(\Omega)^{1/p} (\sup_{x \in \Omega}|f(x)|) < \infty. $$

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