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.
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.