# Inclusion of $C^0(\bar\Omega)$ in $L^p(\Omega)$

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 $$p then $$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.

• 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. Feb 5, 2020 at 10:20
• You are not missing anything. These facts are quite trivial and your arguments are correct. Feb 5, 2020 at 10:21
• Thanks guys, I thought it was rather simple but I preferred to put things written down also to clarify all the passages :) Feb 5, 2020 at 10:23

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