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I understand an $L^p$ function have in them as a dense subset the set of functions with compact support. But do there exist $L^p$ functions that do not have compact support. What are some examples for elementary cases like $p=1,\ldots$?

I hope I am not going in circles here, but I think my question is sensible. Although rationals are dense in reals, the two are still distinct. In the same way, although functions with compact support are dense in $L^p$ space, there should exist $L^p$ functions without compact support. Thanks!

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    $x \mapsto e^{-x^2}$2011-11-08
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    Is support actually a well-defined notion for elements in $L^p(\mathbb R)$? Is there such a thing as an essential support?2011-11-08

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