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Let $M$ be a closed subspace of $L^2([0; 1]; m)$ that is contained in $C([0; 1])$, where $m$ denotes Lebesgue measure. I want to prove that there exists some positive number $K$ such that $||f||_{sup} \leq K ||f||_2$ for all $f \in M.$

I think the Closed Graph Theorem can be applied to solve this problem. I have spent almost two hours, unfortunately, so far but I don't see any clue: How I can use the Closed Graph Theorem? Most importantly, I could not find a closed linear map. Could you give me some hints/suggestions? Any help will be highly appreciated. Thanks so much.

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You have to show that that the inclusion map from $M$ into $C[0,1]$ is continuous. Suppose $f_n \to f$ in $L^{2}[0,1]$ and $f_n \to g$ in $C[0,1]$ . We have to show that $f=g$. Since $f_n \to f$ in $L^{2}[0,1]$ there is a subsequence which converges to $f$ almost everywhere. It follow that $f=g$ almost everywhere. But $f$ and $g$ are continuous. Hence $f(x)=g(x)$ for all $x$. This proves that the inclusion map has closed graphs, as required.

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  • $\begingroup$ Could you please explain a little more? How the closeness of the inclusion map yields the result? $\endgroup$ – James Apr 4 at 4:59
  • $\begingroup$ @James It is given that $M$ is closed. The inclusion map is a linear map between Banach spaces. Once we show that the graphs is closed it follows that it is bounded. The inequality we are asked to prove is exactly boundedness of this map. $\endgroup$ – Kavi Rama Murthy Apr 4 at 5:05
  • $\begingroup$ Thank you so much. Could you please recommend me a couple of simple example like this one related to the Closed Graph Theorem so that I can practice? Or, some materials where I find some straightforward examples/applications of this theorem. Thank you again. $\endgroup$ – James Apr 4 at 5:12

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