# Give and example of a function such that f \in L^2(R) and f \notin L^1(R) [closed]

Let L^2(R)={f:R->C| \int_{\infty}^{\infty} \abs(f(t))^2 < \infty } and L^1(R)={f:R->C| \int_{\infty}^{\infty} \abs(f(t)) < \infty }. Give and example of a function such that f \in L^2(R) and f \notin L^1(R).

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## closed as not a real question by muad, Chandru1, Ｊ. Ｍ., Carl Mummert, RasmusOct 24 '10 at 12:33

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Hint: it is a theorem that on compact subsets of R, the $L^2$ norm is "stronger" in the sense that an $L^2$ function on a compact set much be also in $L^1$. So any example will involve behaviour at $\infty$. –  Willie Wong Oct 24 '10 at 12:24
This one I disagree with the closure. I think the question is clear. It does sound like a standard homework and I think Jonas' answer is a good hint. –  Ross Millikan Oct 24 '10 at 15:19
@Ross Millikan: I agree, it is a very natural question indeed. –  AD. Oct 25 '10 at 4:44
Try to look at a function of the form $f(x) = x^{-a}$ for some $a > 0$ on an appropriate domain.