The following problem is a stumbling block in a research project that I am working on:
Problem. Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it true that $ {C_{c}}(G) $ is a meager subset of $ {L^{2}}(G) $ with respect to the $ L^{2} $-norm topology?
Thank you very much!
Yes. Since $G$ is $\sigma$-compact it is enough to show that $X_K= \lbrace f\in C(G):$ supp$(f)\subseteq K\rbrace$ is meager (of first category) in $L^2(G)$. Otherwise, the inclusion map $i_K: X_K \to L^2(G)$ between Banach spaces (of course, $X_K$ endowed with the sup-norm) would have a range of second category and the open mapping theorem (in the version stated e.g. in Rudin's functional analysis book, theorem 2.11) implies that $i_K$ is an isomorphism. This is the so only in very trivial cases (probably only finite groups).
