ca.analysis and odes - What's the space of smooth functions in L^2(R)?

N.B. this answer was in response to an earlier version of the question, which only had the first two paragraphs -- hence it doesn't address what appears to have been the original poster's actual question. For that, see the answers of Leonid or Harald.

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I'm not sure if this answers your question, but it might be worth noting that a measurable function $f$ on the real line is in $L^2({mathbb R})$ if and only if its Fourier transform $widehat{f}$ is (Plancherel theorem), while it is in $C^infty({mathbb R})$ if and only if we have

$$ int_{-infty}^infty | widehat{f}(x) |^2 (1+ |x|^2)^{k} < infty quad{rm for } k=1,2,dots $$

(this is a form of Sobolev embedding, albeit in a very special case). In particular, if I've correctly understood the notation from the wikipedia page for Sobolev spaces, the space you're after seems to be the intersection $bigcap_{k=0}^infty H^k({mathbb R})$. I don't know if this goes by a particular name.