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

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

1 Answer

Try to look at a function of the form $f(x) = x^{-a}$ for some $a > 0$ on an appropriate domain.

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This looks like a homework problem to me. –  Carl Mummert Oct 24 '10 at 11:57
    
Maybe you're right but I cannot judge that just from the question. I will edit it a bit. –  Jonas Teuwen Oct 24 '10 at 12:02
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In this case, you can also look at the list of the questions by the same editor. There has been someone asking homework questions about functional analysis for a few days. –  Carl Mummert Oct 24 '10 at 12:08
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