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Give an example of a function which belongs to L2 which is not at L5-LOC. The domain is 0 to infinity. I thought about $$ {x}^{-5} $$ . what do you think ?

sheep
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  • $\left(\frac{1}{x^5}\right)^5 = \frac{1}{x^{25}}$. $f^5$ is still integrable around $\infty$, as $f^2$ is; neither $f^5$ nor $f^2$ are integrable around $0$ (i.e., $f$ is not in $L_2$ anyway). – Clement C. May 14 '14 at 14:31
  • Surely you know the definition of the spaces involved? Why not test your guess yourself? – Nate Eldredge May 14 '14 at 14:51

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You could use a variation of the (quite usual) "triangle construction" tailored to $L_2$ and $L_5$ instead of $L_1$ and $L_2$ -- see this post (you can also adapt the first function in Edwin's comment (comment to the original question) to your question, with the issue now being around $0$ and not $\infty$).

Community
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Clement C.
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