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).
Give and example of a function such that f \in L^2(R) and f \notin L^1(R)
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real-analysis
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1Hint: 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 $\infty$. – 2010-10-24
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3This 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. – 2010-10-24
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0@Ross Millikan: I agree, it is a very natural question indeed. – 2010-10-25
1 Answers
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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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1This looks like a homework problem to me. – 2010-10-24
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0Maybe you're right but I cannot judge that just from the question. I will edit it a bit. – 2010-10-24
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1In 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. – 2010-10-24