The following problem is a stumbling block in a research project that I am working on:



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> **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?
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Thank you very much!



Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact Hausdorff group $ G $?
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). 


